Wine with sediment and cases of the like: One who filters dregs [of wine], meaning that he filters [wine which contains dregs] through a strainer, [has done an] offshoot of [the] separating [prohibition which is to separate] food from its waste using a sifter and sieve, or [has done] an offshoot of [the] Sifting/Miraked [prohibition] and is liable. [See Q&A]
Filtering wine with sediment by placing twigs into the opening of the bottle: A vessel into which one empties the wine from the barrel may not have strongly inserted into its opening twigs and reeds, as this is similar to a filter.
The reason: Now, although it’s not an actual filter being that dregs still pass through it, nevertheless since there are twigs and dirt which do not bypass it is thus similar to a filter.
However if [in the barrel] there is clear wine which does not contain sediment and rather only thin splinters, then it is permitted to filter it even using an actual strainer as will be explained next.
Clear wine and clear water in a strainer: Water or wine which are clear are permitted to be filtered through a filter in order so they become even more clear, or even in order [to remove] the sediment that floats on top of the wine or [to filter out] small twigs that have fallen into it. [See Q&A ]
The reason that this is allowed: This is not considered like separating being that [the wine and water] are still drinkable to majority of people even without this filtration. [See Q&A]
The reason why there is no whitening prohibition involved by a strainer: Since a strainer is made for this purpose of having liquids pass through it thus one does not care to squeeze [out the liquid which it absorbs] and as well does not care to whiten it at all, therefore its whitening does not pose a problem to prohibit soaking it in liquid due to this even according to those opinions which say that the soaking of a cloth is [considered] laundering it.
Filtering water through a filter which contains dregs of wine: If one had placed dregs in the strainer from before Shabbos then it is permitted to pour water over it in order so the water become more [clean and] clear. [See Q&A on tea]
Filtering pre-fermented wine:  Wine that has just been pressed, so long as it remains within its primary fermentation [process] one may mix a barrel of it together with its dregs and may place it even inside a strainer.
The reason for this is because: so long as the wine is still within its [primary] fermentation the wine has not yet separated well from its dregs and thus the entire [mixture of the] wine [with its dregs] is considered one food, and there is [thus] no [prohibition] of separating food from its waste involved here.
Furthermore [there is no problem of separating here because] the wine is fit to be drunk even without filtration, and this is commonly done during the times of its pressing when all wines are murky and majority of people drink it the way it is without filtering it.
Filtering vinegar:  The same applies with vinegar, which is likewise common to use even when it is still slightly thick, that it is permitted to filter it. However if [the vinegar] is very thick and is not fit at all [to be used] without filtering then it is forbidden to filter it even with a cloth.
Filtering almond milk: Similarly the milk of crushed almonds [which were made] before Shabbos is permitted to be filtered being that it is able to be drunk without filtering.
Placing milk curd into a strainer: If one places the curd into an elastic vessel and the whey drips from inside it then this is an offshoot of separating food from waste through using a sieve and sifter and one is thus liable.
The reason: Now, although the curd and the milk are both a single species of food and there is [thus] no waste [here] at all, nevertheless since they are liquid substances which mix well which he is [now] separating, therefore this contains the separating [prohibition] just like one who is separating food from waste.
If the liquid is drinkable in its current state to majority of people, then it is allowed to be filtered, [even if it contains pieces of food or dregs such as almond milk and wine that is in its first stage of fermentation]. If it is not drinkable to majority of people in its current state then it is Biblically forbidden to filter.
Placing a filter on the lid of a bottle and pouring through it:
Has all the filtering restrictions mentioned by a filter even if it does not help to filter out all of the sediment.
May one filter murky liquids, which are not drinkable to majority of people, for the purpose of washing dishes and the like?
Even though that for washing dishes the water is perfectly useable even without being filtered, nevertheless it is forbidden to filter it in a filter. However if the water is mixed with soap or other ingredient which has made it no longer drinkable, then it may be filtered so long as it is fit for washing with even without filtering it.
May one today filter wine that contains small twigs even though today such an occurrence is uncommon?
No. Being that today such an occurrence is uncommon, most likely people would be particular to filter it out and thus doing so would be forbidden on Shabbos.
May one filter cork pieces out from wine on Shabbos?
Using a strainer to filter it: Depends. If the pieces are small in size and in amount, then majority of people are not particular to drink it in an unfiltered state, and it is allowed to be filtered if the person himself is also not particular against ever drinking the wine together with these small pieces. However if the pieces are large, or are many, or one is particular to never drink wine with cork pieces, then it is forbidden to be filtered.
Removing the pieces with one’s hand: In all cases that it is forbidden to filter the pieces it is forbidden to remove them with one’s hands.
Regarding cases that it is permitted to filter the wine then: There are opinions which forbid doing so. Others however permit this to be done being that anyways people drink it in this state, and thus it is not considered separating bad from good.
Pouring the cork pieces out together with some wine: Is permitted according to all.
May one who is particular to never drink unfiltered water filter it on Shabbos if it is drinkable by majority of people?
If the water is slightly recognizably dirty, but nevertheless majority of people will drink from it then: It is nevertheless forbidden for one who is always particular, to filter it on Shabbos. [See next two questions]
If the water is clean: and one thus simply desires to make it clearer then it is allowed for one to filter it even if he is always particular against drinking it. Nevertheless there are opinions which are stringent even in such a case to prohibit a person which is always particular from filtering it. Thus according to those opinions people which are particular to only drink filtered water must prepare from before Shabbos water for them to drink on Shabbos.
If one does not yet know the state of the water, such as water which enters directly from the tab into a filter, then it has the same status as clear water.
May a person which is not particular against drinking unfiltered water filter water for one who is particular in doing so and may thus not filter it?
There is an opinion which leans to be lenient to allow one to filter for his particular friend although he concludes that he has not relied on this viewpoint to actually allow it to be done. [Thus in pressing situations, one should seek a Rav to see if one can rely on this side for leniency.]
May one who is particular to only drink filtered water, filter water for one who is not particular to do so?
One may not be lenient to do so.
May one use a Brita filter on Shabbos to filter water?
If majority of people drink this water in its current unfiltered state, and the person which desires to filter it also does not mind to drink it in its current unfiltered state and just wants to filter it so it be even more clean from any possible waste, then it may be filtered. Regarding if the person who wishes to filter it is particular to never drink this water in its unfiltered state- see Q 4 above.
If majority of people filter this water due to a problem with water, then it is always forbidden to filter it on Shabbos. Thus water that has been infected with bugs or contaminated with sand and the like which may only be drunk after being filtered, it may not be filtered on Shabbos and one must thus prepare from before Shabbos enough water for Shabbos.
Regarding if the water is clear and no change is seen in the water through filtering-See Q&A above!
May one use the tab filter contained in all faucets on Shabbos or does this pose a filtering prohibition and thus must be removed before Shabbos?
Yes one may use it on Shabbos being that the water is fit to be drunk by majority of people in its current state without being filtered [see footnote for additional reason]. Regarding if the person who wishes to filter it is particular to never drink this water in its unfiltered state- see Q 4 above and footnote below.
Regarding if the water is not fit for majority of people [and will become fit through passing through this filter]- see above mentioned footnotes that according to the Minchas Yitzchak this would be allowed. See also the next question.
If the only available water is undrinkable, such as it contains sand or worms and thus must be filtered, is there any way to filter it on Shabbos?
It is forbidden to take the water and pour it through any type of filter unless one is doing so in the process of drinking, as will be explained in Halacha 4-See there.
Using a filter that is attached to the sink: However one may use a filter that is attached to the spout of the faucet and thus have the water automatically get filtered as it leaves the faucet. [This however does not include filters which have a tube leading into the spout of the faucet of which one has an option to ether have the water come through the filter or come through the faucet, as in such a case the filter is only used for drinking water which negates the basis for the allowance of a sink attached filter, as explained in the footnotes of the previous question.]
Even when attached to the spout, ideally it is best not to use a non-cloth filter [see footnote] and rather a designated cloth filter is to be used.
May one pour a liquid which contains solids into the drain of the sink if the drain has a filter [as do most sink drains]?
If both the liquid and solid are waste: Such as leftovers, then this is allowed.
If the solid is not waste: Such as that a ring fell into a mixture and one desires to find the ring by placing down this filter of the drain, then it is forbidden.
May one use a spoon with holes to remove a solid out from a liquid, such as vegetables from soup?
As already explained in Chapter 1 Halacha 4 Q&A there, in all cases that the solid and liquid are designated as mixed [see Introduction to chapter 1 and Q&A there] it is forbidden to remove any solid from the liquid using a straining spoon being that doing so is considered separating with a vessel which is Biblically forbidden. With regards to if one desires to eat both the liquid and solid right away but separately see Halacha 6 Q&A there.
May one filter fruit pulp out from ones fruit drink?
If majority of people drink this juice together with its pulp, and the person who desires to filter it also does not usually mind to drink it together with its pulp, then it may be filtered.
However if the person who wishes to filter it is particular to never drink this juice without first filtering the pulp or even if he is not particular but majority of people filter out the pulp, then it is always forbidden to filter it on Shabbos. Thus such juice one is not allowed to pour into a cup through a pitcher which has a screen on its top which withholds the pulp.
For a baby: According to the above it is forbidden to filter the pulp out of juice in order to give to a baby to drink if one is filtering it due to choking hazard for the baby, as the baby is considered like a person which is always particular to filter the juice. However there is no problem to filter it simply in order to prevent the bottle hole from getting stuffed.
May one filter baby cereal from their lumpy parts? 
If majority of people are not particular to filter out the lumps, and the person who desires to filter it also does not usually mind to drink it together with the lumps, then it may be filtered. If the person is particular then it is best for him to filter it using a cloth rather than a standard filter even though the latter is also valid from the letter of the law.
For a baby: Some Poskim hold that so long as the lumps are able to be eaten by an older baby, then one may filter it for any age baby. However even so it is best to do so through a cloth. There is no problem to filter it in order to prevent the bottle hole from getting stuffed.
May one filter the fat out from milk? 
The layer of fat which covers the top of the milk: Is forbidden to be filtered being that most people today do not drink the milk with this fat.
Small pieces of fat: If one is not always particular to filter it then it has the same law as slightly murky liquids which may not be filtered through a filter but may be filtered with a cloth- see Halacha 2 Summary.
If one is always particular to filter it then he should be stringent.
Q&A regarding tea bags:
May one use a tea pot that has a strainer on its tip to prevent the leaves from falling in the cup?
It is permitted to pour the tea from this pot until one reaches the liquid on the bottom of the pot where the tea is mixed with the tea leaves. Regarding if one may pour out the tea past this point many Poskim allow doing so. [This applies even according to the ruling of Admur in the Siddur which views that the food which is left in the pot is considered that which is separated, and thus should ideally be prohibited]. However others rule that one is to be stringent.
In any event when relying on the lenient opinions he should only do so for right away use.
May one use tea bags on Shabbos?
Yes. This does not pose a filtering prohibition, despite the fact that while the bag is in the liquid the liquid is being filtered through the herbs in the bag, absorbing the taste of the tea bag.
May one shake the tea bag while inside the liquid in order to dissipate more tea essence?
May one remove a sliced lemon from one’s tea?
Yes as large items are not defined as “mixed” with liquid. Regarding if initially it is allowed to place lemon into liquid on Shabbos, see “The laws of Cooking” chapter 1 towards the end.
May one use his hand to strain food for right away use?
No as this is not considered the way of eating, and it is thus equivalent to using a strainer. However see Halacha 6 below and the footnote to the Q&A there.
 Today this permission no longer applies being that majority of people today are particular not to drink wine with twigs until it is filtered, being that today it is very uncommon for wine to include twigs. [Ketzos Hashulchan 125 footnote 28. Shabbos Kehalacha Vol 2 page 348]
 With red wines, the must is pressed after the primary fermentation, which separates the skins and other solid matter from the liquid. Thus during the primary fermentation the dregs still remain with the wine.
 This refers to a vessel made of material that can be stretched out do to having tiny holes in it, and thus allows the milk to drip out. Similar to a plastic bag.
 319/16. Shabbos Kihalacha Vol 2 page 345
 Shabbos Kihalacha Vol. 2 p. 349
 Shabbos Kehalacha Vol. 2 p. 348
 Ketzos Hashulchan 125 footnote 28
 Shabbos Kehalacha Vol 2 page 348-349
 Ketzos Hashulchan chapter 125 footnote 28, [although there he concludes with a Tzaruch Iyun due to that he brings also the opinions which allow this]. Tehila Ledavid, and Rav Farkash in Shabbos Kehalacha Vol 2 page 348-349.
 As the laws of filtering are different than that of separating, as by filtering it is not clearly evident that one desires to separate the pieces as perhaps he just wants clearer wine, while when separating the pieces with ones hands this is clearly evident, and is forbidden as here one is clearly separating the bad from the good.
 Shevisas Shabbos and Tal Oaros.
 Shabbos Kehalacha Vol. 2 p. 347 and 361
 So rules Peri Megadim 319 M”Z 6
 Based on ruling of Rav SZ”A regarding Pesach cloth, and Minchas Yitzchak 7/23, Ketzos Hashulchan and others that by waste that is not recognizable the separating prohibition does not apply. So is also implied from the Lashon of the Peri Megadim which is the source for the stringency which mentions “water that has small twigs” in his case of a particular type of person.
 Ketzos Hashulchan 125 footnote 37
 Shabbos Kehalacha page Vol. 2 347
 Shut Maharahm Shik
 Shabbos Kehalacha Vol 2 page 347
 Rav SZ”A questions whether this is allowed or not and concludes with a Tzaruch Iyun.
 Shabbos Kehalacha Vol. 2 P. 346-347
 So rules the Peri Megadim in M”Z 6
 Shabbos Kehalacha Vol. 2 p. 363
 This is in addition to the allowance of the Minchas Yitzchak [7/23] regarding using a cloth that is attached to the faucet being that since one uses the cloth for all waters which come from the sink, including washing and the like, therefore even the water used for drinking is not considered problematic. To note however that not all accept this argument [See Ketzos Hashulchan 125/37 end from which it is proven that he did not hold of this Heter]. The ramification between this reason and the one mentioned above is in a case that in truth the water is not fit to be drunk by majority of people, in which according to the Minchas Yitzchak one would be able to use it. See next note.
 According to the Minchas Yitzchak mentioned in the previous footnote this would be allowed regardless of what was explained in Q 4 being that the filter placed in the faucet is used for also washing and the like.
 Shabbos Kehalacha Vol. 2 p. 362
 So writes Minchas Yitzchak, and so concludes Rav Farkash, in order so one do the filtering with an irregularity. Nevertheless from the letter of the law the Minchas Yitzchak rules that even a metal filter is allowed when attached to the spout of the faucet being that it is also filtering the non-drinking water.
 Shabbos Kehalacha Vol. 2 p. 362
 So rules Rav SZ”A [SSH”K 12 footnote 47], Igros Moshe [4/74-4] Tzitz Eliezer [7/12]
 Shabbos Kihalacha Vol. 2 p. 375
 So rules the Peri Megadim in M”Z 6
 Shabbos Kihalacha Vol. 2 p. 376
 So rules the Peri Megadim in M”Z 6
 Shabbos Kihalacha Vol. 2 p. 377
 Shabbos Kihalacha Vol. 2 p. 368
 As is always the rule regarding pouring one thing from another, as the fact that there is a strainer on the top of the pot does not make a difference in this regard.
 The problems faced in doing so is that a) one is separating with a vessel [the strainer that is attached to the top of the kettle] and b) According to Admur in Siddur one is separating the bad from the good.
 So rules Shevisas Shabbos, Chazon Ish, Ketzos Hashulchan 125 footnote 21, Rav SZ”A.
The problems mentioned in the previous footnote are answered in the following ways according to the lenient opinions: a) The tea leaves are fit to be drank together with the liquid [by most people] and thus it is permitted to strain the liquid from it, as is explained in the laws of straining. [Minchas Shabbos/ Ketzos Hashulchan] b) One does not intend to separate the leaves from the tea but rather just to drink the tea. [Ketzos Hashulchan] c) A vessel which is designated to separate for only right away use, as is a kettle, is not included in the separating restrictions. [Rav SZ”A in Shulchan Shlomo].
 So is proven from the fact that the Ketzos Hashulchan which rules like Admur in the Siddur rules that it is allowed in our case for the reasons mentioned in the previous footnote.
 Sheivet Haleivy, and others brought in Kaf Hachayim 113.
 So rules Rav Farkash, as then one can also join in the reason of Rav SZ”A mentioned in the previous footnotes.
 Shabbos Kehalacha Vol. 2 p. 370
 So rules Minchas Yitzchak 4/99-2
This is because this is similar to the case of filtering water through a filter that contains dregs of wine which is allowed despite the fact that the water is mixing in with the dregs and then becoming filtered out.
 Shabbos Kehalacha Vol. 2 p. 370
 Seemingly according to the Minchas Yitzchak [ibid] this would be allowed. However others have argued on this -See Shabbos Kihalacha ibid
 Thus relying also on the allowance of Rav SZ”A that any vessel used for only right away use does not contain a filtering prohibition.
 Shabbos Kehalacha Vol. 2 p. 374
 Shabbos Kehalacha Vol. 2 p. 381