Filtering the coffee liquid from the ground powder:
The general rule: The Borer prohibition of filtering liquids only applies towards liquids that contain solids which people are not accustomed to eating. However, if the solid material that is found in the liquid is consumable by majority of people together with the liquid, then the liquid may be filtered. This, however, only applies to one who is not always particular to filter the liquid from the solid. If, however, one is always particular to avoid drinking the liquid with this solid, then it is forbidden for him to filter it on Shabbos even if it is drinkable by majority of people. Practically, many people [especially in the army] drink their coffee drink together with the ground coffee [i.e. cowboy coffee, mud, Turkish coffee] and are not particular to filter it. Other’s however are particular to always filter their ground coffee, and it is unclear as to which group is in the majority. Using this information, we will now analyze the different forms of filtration available and their practical law:
Pouring the coffee liquid into another cup, having the ground coffee remain on bottom of the original cup: It is permitted to pour the coffee liquid out from the cup without any restriction so long as one leaves some amount of liquid remaining above the ground coffee that remains on bottom. To pour coffee past this point, and completely separate all the coffee liquid from the ground coffee is permitted according to most Poskim if done to drink right away. However, according to Admur in his Siddur this matter is under several grounds of debate even if one intends to drink the coffee right away and is hence better to be avoided. However, those who are lenient have upon whom to rely especially if they will be drinking the coffee right away and are not always particular to filter it.
Pouring the coffee using a French press strainer and the like: It is permitted to pour the coffee liquid out from the French press without any restriction, so long as one leaves some amount of liquid remaining above the strainer that rests on bottom. To pour coffee past this point, and completely separate all the coffee liquid from the ground coffee which is below the strainer, is debated amongst the Poskim. Some Poskim rule it is forbidden to do so, as coffee powder is not edible by majority of people, and it is forbidden to separate using a strainer. Other Poskim, however, justify the custom to be lenient to allow filtering the coffee liquid even from the bottom area of the strainer, under several grounds. Practically, it is best to avoid pouring out all the liquid and one should rather stop pouring as soon as one reaches the area of liquid that is above the strainer/coffee powder, in which case it is permitted according to all. This especially applies to one who is always particular to filter his ground coffee, in which case he is certainly to avoid pouring out all the coffee liquid.
Pouring hot water into a coffee strainer which contains ground coffee: It is permitted to place ground coffee into a strainer and then pour water [from a Keli Sheiyni or onwards] onto it, thus having the water pass through the ground coffee and strainer. [This applies even if the water will take some time to pass through the strainer. However, one may not pour coffee liquid that is already mixed with the ground coffee through a strainer in order to strain the ground coffee. This especially applies if one is always particular to strain it, in which case it is certainly forbidden to do so, as stated above. Likewise, even when pouring the water onto the ground coffee which is in the strainer, one is to beware of the following: a) Not to do any action to help the water strain faster or better, such as not to shake it, mix it using a finger or a vessel. B) If the strainer is sitting inside the liquid, one is to remove it using a spoon and the like to avoid any drops from falling into the coffee after it is removed.]
It is permitted to make a hot beverage of ground/Turkish coffee on Shabbos through using a Keli Shelishi or Iruiy Keli Sheiyni.
Pouring from a cup and French press: One may likewise filter the coffee liquid from the ground coffee powder through pouring the liquid into another cup, although making sure to stop pouring as soon as one reaches the area in which the liquid is mixed with the coffee powder. This allowance applies even if one is using a French press. However, it is best not pour the coffee liquid out once it reaches the level area of the ground coffee/strainer, even if one plans to drink it right away.
Pouring through a strainer: Certainly, one should not pour the coffee liquid directly through a strainer in order to filter the ground coffee. One may however pour hot Keli Sheiyni water into a strainer that contains ground coffee and have the liquid drip into one’s cup. One, however, should beware not to shake the strainer and is to remove it from the liquid using a spoon.
 Admur 319:13; Ran; Rashba; Beis Yosef; first opinion in M”B 319:34; Tzemach Tzedek Yoreh Deah 45; Shabbos Kehalacha 2 p. 346
Other opinions: Some Poskim rule it must be fit for drinking to all people. [Bach brought in M”B 319:10; See Tzemach Tzedek ibid]
 See Admur 319:13 “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. This is not considered like separating being that [the wine and water] are still drinkable to majority of people even without this filtration”; Rama 319:10 “because it is fit to drink regardless of the twigs”; Michaber 319:9; Shabbos 139b
Other opinions: Some Poskim rule one may never use a filter on Shabbos even if the liquid is clear and fit to be drunk by majority of people. [Rambam 8, brought in Michaber 319:10] Practically, we do not rule like this opinion. [M”B 319:41; Admur omitted this opinion]
 See Peri Megadim 319 M”Z 6, brought in Ketzos Hashulchan 125 footnote 28 and Biur Halacha 319 “Hoil”; Maharam Shick 134; Shabbos Kehalacha Vol. 2 p. 347 and 361 footnote 52; Piskeiy Teshuvos 319:33
 Admur 319:18; “It is permitted to gently pour [food or liquid] from one vessel into another in order so the residue and dregs remain on the bottom of the vessel [being poured from]. However, one must be careful once the stream [being poured] has stopped and small trickles begin to drip out from the last remainder [of food that is] amongst the waste, then one must stop [pouring] and leave [that remainder of food] together with the waste. The reason for this is because if one were to not do so, then these last drops prove [to all] that [his true intentions in this pouring] were to separate [the food from the waste]. However, during the initial pouring when the waste is not yet recognizable, then he has done no separation.”; Michaber 319:14; Kaf Hachaim 319:113 regarding tea and all mixtures
 Admur 319:18; M”A 319:15; M”B 319:55; Kaf Hachaim 319:113
 The reason: As a) perhaps coffee is not considered drunk by majority of people with its ground powder and hence contains the Borer restrictions. [Chayeh Adam 16, brought in Ketzos Hashulchan 125 footnote 21] b) According to many Poskim, Admur in his Siddur rules that when one pours liquid from a cup it is considered that one is removing the solid which remains in the cup from the light, and if forbidden due to the Borer restriction against removing bad from good. [see next footnote]
 Arguments to permit: 1) Perhaps coffee is considered edible by majority of people together with its ground powder, and hence has no filtering/Borer prohibition. [Ketzos Hashulchan 125 footnote 21] 2) If he pours to drink the coffee right away, many Poskim rule it is considered like taking the food from the waste which is permitted for right away use. [Admur 319:18; M”A 319:15; M”B 319:55; Kaf Hachaim 319:113] Now, although Admur in the Siddur seemingly retracted from this view [Divreiy Nechemia 21; Ketzos Hashulchan 125:9 and footnote 21; Iglei Tal Borer 5; Chayeh Adam 16:9, brought in Shaar Hatziyon 319:44; Oar LeTziyon 2:31-10.], those who are lenient, especially in light of the first argument, have upon whom to rely. [See Shabbos Kehalacha Vol 2 pages 113-122]
 See Shabbos Kihalacha Vol. 2 p. 368; Piskeiy Teshuvos 319:37
 See sources previous footnotes
 Chayeh Adam 16, brought in Ketzos Hashulchan 125 footnote 21; Beis Menucha 48; Ben Ish Chaiy Beshalach 2:18; Kaf Hachaim 319:113; Shevet Halevy 1:84; Bris Olam Borer 40; Az Nidbaru 1:23; See Piskeiy Teshuvos 319:37
 Shevisas Hashabbos Borer footnote 29 and 49 regarding tea questions the allowance but then concludes that so is the custom; Ketzos Hashulchan 125 footnote 21 regarding also coffee; Chazon Ish 53; Rav SZ”A in SSH”K 3 footnote 125; See Minchas Yitzchak 7:23; Or Letziyon 1:27; Yechaveh Daas 2:51; Piskeiy Teshuvos 319:37
 The reason: As a) As [most] people are not particular against consuming some of the ground coffee together with the liquid and it is thus permitted to strain the liquid from it. [Shevisas Hashabbos ibid; Ketzos Hashulchan ibid] b) In any even the coffee powder goes to the bottom and is hence not considered to be strained. [Ketzos Hashulchan ibid] c) One does not intend to separate the coffee liquid from the coffee powder but rather just to drink the coffee. [Ketzos Hashulchan ibid; Chazon Ish ibid] d) 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 SSH”K ibid].
 See Peri Megadim 319 M”Z 6; brought in Ketzos Hashulchan 125 footnote 28; Shabbos Kehalacha Vol. 2 p. 347 and 361 footnote 52
 See Admur 319:12, Michaber 319:9 and Mishneh Shabbos 139b “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.”; Shabbos Kehalalacha 14:36 and Biurim 8; SSH”K 3:64; Piskeiy Teshuvos 319:32 that from here we can learn the allowance to pour water onto coffee that is in a strainer; Minchas Yitzchak 4:99-2 and Shabbos Kehalacha Vol. 2 p. 370
Must the coffee be placed in the strainer on Shabbos? No. [See Piskeiy Teshuvos ibid footnote 274]
Is there a prohibition of Ohel in covering the vessel with the strainer: No. [See Admur Admur 315:19-20; Piskeiy Teshuvos ibid]
 Pashut. The Poskim do not differentiate in this matter, and this is in fact the Yesod for allowing using tea bags on Shabbos. [See Shabbos Kehalaha ibid in Biurim]
 This case is even more severe than the case of a French press, as the only logic of allowance that remains in this case is that coffee powder is edible to majority of people, which is under debate.
 Minchas Yitzchak 4:99; Shevet Halevi 8:58; Piskeiy Teshuvos ibid; See however See Shabbos Kehalacha 2 p. 370
 Minchas Yitzchak 4:99-2; Sheivet Halevy 8:58; SSH”K 3 footnote 171 in name of Rav SZ”A; See however Shabbos Kehalacha Vol. 2 p. 371 based on SSH”K 3 footnote 125