The Biblical Prohibition of Separating:
A. The General law:
Intro: It is Biblically forbidden to separate any mixture which was defined above as a mixture by either a) removing the bad from the good, or b) using an instrument designated for separating or c) doing so for a later use.
Removing the waste from the food: Separating is one of the principal [Shabbos prohibited] actions, as in the Tabernacle they would separate the waste from the herbs [used for the dyes] and therefore anyone who separates waste from food even with one’s hand and even in order to eat right away, is liable.
The reason for why removing the waste is forbidden: The removing of the waste [from the food] is not [considered done in] the normal framework of eating [which is permitted to be done as will be explained below] but is rather [considered] preparing the food so it be fit to eat which is a complete [Biblically prohibited] act. Therefore even if [the mixture contains] a lot more food than waste and there is greater bother in having to separate the food, nevertheless one may not separate the waste even in order to eat [the food] straight away.
Removing the food from the waste- Using a Sieve and Sifter: As well, one who separates food from waste with a sieve or sifter is Biblically liable even if one did so in order to eat the food right away.
The reason for this is: because using a sieve or sifter is the common form of sifting food [and is thus not considered a form of eating but rather a form of preparation].
Separating the food in order to eat later on: As well one who separates food from waste without intent to eat it right away but rather to leave it to be eaten later on, even [later on] that day, then it is [considered] like he had separated [the food to place it in] storage and he is liable.
The reason for this is: because it is not applicable here to say that what he did was within the context of eating being that he is not eating it right away.
Summary-The Biblical prohibition:
It is Biblically forbidden to separate in any one of the following ways:
1. Bad from good
2. For later use.
3. With a utensil designated for separating.
B. May one remove the bad together with some of the good thus leaving the remainder of the mixture free of the bad?
The opinion of Admur in the Shulchan Aruch:
It is permitted to remove from food which has waste in it, the waste together with some of the food, such as [using a spoon to] remove a fly from a cup together with some of the liquid, being that then it does not appear like one is separating at all. [However see footnote for ruling of Peri Megadim].
The opinion of Admur in the Siddur:
One may not rely on the customary permission granted that if there is waste in ones food then one removes it together with some of the food etc, as doing so is questionable of containing a sin offering liability and a prohibition of Sekilah G-d forbid. [See Q&A regarding if this applies even when one leaves some of the bad still with the good.]
It is forbidden to separate the bad even when one takes with it some of the good.
If one will separate only some of the bad together with good and will thus still leave some of the bad with the mixture do the Borer regulations still apply?
This is Rabinically forbidden to be done in all cases that separating would involve a Chatas obligation.
C. May one blow away waste from amongst food?
- The Siddur:
One may not blow on [the waste] with his mouth until it is blown out of the food. [See Q&A]
May one blow the nut shells and chaff away from a grouping of opened nuts, as is common with opened peanuts which have their chaff [the brown thin shelling] mixed with them?
This is forbidden being that blowing away the bad is considered like separating it with ones hands.
D. May one soak food with its waste in water in order to separate and remove the waste:
Food and waste which are mixed [together] is forbidden to be placed into water in order for the waste to sink to the bottom, as is the case if the waste is earth, or [so the waste] float to the top as is the case [if the waste] is straw.
Similarly Karshinin may not be soaked in water in order so their waste float to the top, and as well one may not rub it with his hands [while it is in the water] so the waste float up as doing so is similar to [the] separating [prohibition]. [See Q&A]
However one may place them in a sieve even though the waste [might] fall through the holes of the sieve and it is thus found that [the waste] becomes separated on its own. [One may not however place the Karshinin in the sieve with specific intent that it possibly be filtered from its waste.]
Rinsing off waste from foods: See Q&A
Summary: Separating waste from food through soaking the mixture in water:
Is forbidden to be done due to the separating prohibition [even when done to eat right away].
Q&A on soaking foods in water:
May one soak foods in order to remove waste that majority of people are not particular to wash off?
If the person himself is not always particular about this then he may wash it off in all cases, as the waste is not considered waste.
If the food was already within the water from before Shabbos may one rub the waste off on Shabbos?
No. However see next Q&A!
May one soak a fruit or vegetable in order to remove the pesticide, bugs, or dirt that is stuck on the fruit or vegetable?
If it is not possible to remove the dirt thru simply rinsing the fruit/vegetable then one may pour water over it numerous times until it is clean. If this too does not suffice then one may soak the fruit or vegetable in water. If this too does not suffice one may even scrub them in the water. This allowance especially applies to washing off pesticide which is not even visible. It is however forbidden to soak a group of fruits or vegetables together with their waste as will be explained next. In all cases it is forbidden to wash the fruit or vegetable for later use.
May one soak a basket of small fruits such as blueberries or grapes in water?
If the grapes or blueberries contain stems which will float to the top, doing so is forbidden. If however they are clean of stems and one merely desires to wash off the dirt stuck onto the fruits, then doing so is allowed for right away use if it is not possible to rinse them by pouring water over them.
May one soak lettuce in water?
If it is not possible to clean it of the waste through rinsing then it is permitted to soak it in water for right away use. It is forbidden to add soap to the water, as will be explained below.
Q&A On Washing/Rinsing off waste from food:
May one wash off waste from foods that majority of people are not particular to wash off?
If the person himself is not always particular about this then one may wash it off in all cases, as the waste is not considered waste.
May one rinse away the waste from a mixture of food and waste? 
This has the same laws as soaking them which is forbidden.
May one wash off the pesticide, bugs, or dirt which is stuck on a fruit or vegetable?
One may do so even initially in order to eat right away, as it is considered like peeling off the peel of a food which is allowed to be done in close proximately to the meal. To note however that it is forbidden to wash the bugs off with salt water or vinegar and other liquids which kill the insects, due to the killing prohibition.
May one use soap and the like to wash foods:
Foods which majority of people eat without rinsing then if the person himself is also not always particular to wash this food, it may be washed with soap according to all. However if one is particular to wash it with soap then it is disputed in the Poskim as to whether it is allowed. Thus one should not do so unless there is great need for it.
May one separate the bad from the good using a Shinui/irregularity?
It is forbidden to separate the bad from the good even through using an irregularity, such as through using ones left hand. According to some opinions doing so involves a Biblical transgression, while according to others it is only a Rabbinical transgression.
If one will not separate all the bad from the good or vice versa, and thus will still be left with a mixture of some bad and good do the Borer regulations still apply? 
This is Rabinically forbidden to be done in all cases that separating would involve a Chatas obligation.
What is the law if one accidently did Borer on Shabbos, may the food still be eaten?
It is permitted to be eaten on Shabbos if done accidently or without knowledge of the prohibition.
If one accidently removed the waste from the mixture does he need to replace it back into the mixture?
- One accidentally grabbed a rotten fruit from amongst a pile of fruits.
- One accidentally removed an onion from his salad instead of a tomato.
- One has a pile of blackberries and raspberries and he wants to eat the raspberry and accidently removed a blackberry.
The Law: In a case that the fruit which was taken out is rotten [Example 1] then one does not need to replace the fruit back into the pile. However in a case that the food is fully edible and one simply accidently removed the wrong species of food [Example 2-3], then he must replace the food into the mixture as otherwise this appears like Borer.
May one remove waste from waste, such as to remove peels of Kedushas Shevias away from other waste?
One may only remove waste from waste if he has no intent in using either of the items for any purpose, neither now or later on. Thus one may remove Kedushas Sheviis peels from amongst regular peels in order to place the regular ones in the garbage and the Kedushas Shevias in a separate bag for it to rot.
However if one intends to use one of the items for a purpose then the full Borer restrictions apply.
May one separate for the purpose of verifying whether or not he has a certain item or not?
For example: One desires to eat tomatoes tomorrow by his meal, may he check through his vegetable drawer, removing vegetables, to see if there are tomatoes on the bottom.
Although there are certain Poskim which allow this if finding the object will cause him current satisfaction, nevertheless one should not practically rely on this opinion.
A Chassidic perspective:
The meaning behind Borer: Every food that we eat contains in it Divine sparks of G-dlinesss of which is explained in the Kabalistic and Chassidic teachings to have fallen within the foods during the destruction of the supernal world of Tohu. Through eating foods and then serving G-d with the strength that one benefited from those foods one elevates those Divine sparks back to their source in G-dliness. This service of elevation however only applies during the weekdays. On Shabbos all physicality is elevated to its source and is thus not of need of refinement. This is why Borer is forbidden on Shabbos.
 The curtains and roofing in the Tabernacle were made of dyed skins. The dye was made by boiling different herbs. Prior to the boiling they would separate the mixed waste from amongst the herbs. The Melacha of Borer does not refer to the separating of waste done to the herbs of the incense as only those Melachas which were done in the process of building the Tabernacle are counted as part of the 39 Melachas, and not Melachas done once the Tabernacle was built. [Rav Ovadia Bartenura Mishnayos Shabbos 7/1]
 This is opposed to the ruling of the Tur which holds that in such a case one is allowed to remove the waste rather than the food as doing so is the way of eating. Admur and others however limit this ruling only to Yom Tov and not on Shabbos. [See Shabbos Kihalacha Vol. 2 p. 141]
 319/24, based on Taz 13. So rules also M”B 319/51.
 However from the ruling of Peri Megadim 13 [brought in Q&A] it implies that this Heter to remove the bad with the good only applies in certain scenarios.
 Such as a fly or fat over gravy, one may not remove the fat or fly together with some gravy. [ibid]
 The Ketzos Hashulchan 125 footnote 17 explains that Admur retracted his ruling from the SH”A because since ones main intent is to remove the inedible and the good that is removed is only in order to allow him to remove the fly, therefore the good is nullified to the bad and it is considered that he is removing solely the bad from the good
 Shabbos Kehalacha Vol. 2 p. 93
 Peri Megadim M”Z 13, Aruch Hashulchan 319
 Doing so is prohibited due to the Melacha of Zoreh [winnowing] and not Borer.
 Shabbos Kehalacha Vol. 2 p. 98
 Possible this refers to the horse been which is similar to the broad been and is used for horse fodder.
 So is evident from the wording of Admur “to cause to float” hence implying the Karshinin are currently in the water. If however they are taken out of the water, then rubbing the waste off is forbidden due to the Mifareik prohibition and not due to Borer. In any event it can be derived from here that merely rinsing a food [in contrast to soaking] is not forbidden due to Borer when done with intent to eat right away, while soaking it is forbidden due to Borer. Hence by foods which the Mifareik prohibition is not applicable, one may rinse off their waste with intent to eat the food right away. The reason for the differentiation is because soaking a food is considered Derech Borer while rinsing it is considered Derech Achila. See Q&A! [Ketzos Hashulchan 146 p. 109; Igleiy Tal Borer 16; Shabbos Kehalacha 2 p. 132]
 Some hold that this is not considered the way of eating and is thus forbidden. Others hold that it is like separating the waste from the good.
 Based on 324/1 where Admur writes regarding placing straw in a sieve that it is allowed although “that at times the food falls, as one has no intent to do so and it is not an inevitable occurrence”. So concludes the Ketzos Hashulchan 131 footnote 17, and Shabbos Kihalacha Vol 2 page 356. Thus if it is for certain that the waste will fall through, then it is forbidden. This cannot be proven from Halacha 21 where Admur rules that placing mustard with barn in the strainer is forbidden being that there it is discussing having the bran [waste] remain above and the mustard fall out, which contains a Meraked prohibition, while here it refers to having the waste fall out on its own, which does not contain a Miraked prohibition as will explained in the introduction to the next section.
 324/1 regarding straw. As when one intends to do a certain Melacha, even if it is not inevitable, it is forbidden to be done.
 Shabbos Kehalacha Vol. 2 p. 131; Ketzos Hashulchan 146 footnote 50-10
 This is similar to the law by small pieces of bark in one’s wine which is considered nullified to the wine if one is not particular against drinking it, as will be explained in chapter 3 in the laws of filtering. [Ketzos Hashulchan 146 p. 116] This does not contradict the notion that Borer applies even by two edible foods of different species, as there one cannot say that one species is nullified to another. However here, waste which people pay no attention to is nullified to the food. [Rav SZ”A in SSH”K 3 footnote 156]
 Shabbos Kehalacha Vol. 2 p. 132
 As rules Admur above regarding the Karshinin “and as well one may not rub it with his hands so the waste float up”. See footnote there!
 Ketzos Hashulchan 146 footnote 50-10; Shabbos Kihalacha Vol. 2 p. 134
 As is the case by all small fruits such as grapes and berries. However if the grapes have stems which will float up, then this is forbidden. See next Q&A!
 So rules Ketzos Hashulchan 146 footnote 50-10 as he explains that the above prohibition in truth only applies to the case of Karshini being that if one were to filter it with a sieve it would carry a Biblical prohibition being that it is common to do so, as opposed to the removal of dirt from fruits and the like.
 Ketzos Hashulchan 146 p. 117
 As this is directly similar to the prohibition written above by Admur.
 Shabbos Kehalacha Vol. 2 p. 131
 Shabbos Kehalacha Vol. 2 p. 134
 Shabbos Kehalacha Vol. 2 p. 134-136
 So rules Ketzos Hashulchan 125 footnote 16; 146 p. 116
 Shabbos Kehalacha Vol. 2 p. 199
 Shabbos Kehalacha Vol. 2 p. 136
 The Ketzos Hashulchan [146 footnote 50] debates whether this is allowed and concludes that despite the reasons to allow it he refrains from doing so being that it appears like a belittling of Shabbos and a mundane act. On the other hand the Chazon Ish clearly rules that it is allowed, and so seems to be the opinion of Rav SZ”A. In SSH”K 1 they rule that it is allowed.
 Shabbos Kehalacha Vol. 2 p. 91. Minchas Yitzchak Vol. 5 Chapter 38
 Chayeh Adam brought in M”B 340/22
 Shevisas Hashabbos- Borer.
 Shabbos Kehalacha Vol. 2 p. 93
 Peri Megadim M”Z 13, Aruch Hashulchan
 Ketzos Hashulchan 129 footnote 5; Teshuvos Vehanhagos 2/185
 As although in general we rule that by all Biblical prohibitions done by mistake it is forbidden for all until after Shabbos, nevertheless here being that Borer is so common to be done to food if this were true then it should have been mentioned. As well as that one’s actions is not his source of benefit being that he could have separated it in a permitted way. As well as that here there is no reason to decree Ate Meizid being that one can anyways do so in a permitted way.
 Shabbos Kehalacha Vol. 2 p. 97, and 194
 Shabbos Kehalacha Vol. 2 p. 138
 Shabbos Kehalacha Vol. 2 p. 168
 Shulchan Shlomo