Gemara

109a (Ze'iri): One may strain clear wine or clear water through a Meshameres (a cloth through which wine is strained). Since one could drink it without this, it is permitted.

139b (Mishnah): One may strain wine through a scarf or a wicker basket.

(Ze'iri): One may put clear wine or clear water in a Meshameres on Shabbos without concern, but not if they are cloudy.

Question (Beraisa - R. Shimon ben Gamliel): One may mix up a barrel of wine with its dregs and pour it into a Meshameres on Shabbos.

Answer (Ze'iri): This applies Bein ha'Gitos (when wine is made in the winepress).

Rishonim

Rif (57b): Bein ha'Gitos, when wine is still bubbling and not clear, and it is cloudy, one may mix a barrel of wine with its dregs and pour it into a Meshameres on Shabbos without concern.

Ran (DH Mishnah): Our Mishnah discusses cloudy wine. However, one can drink it. If it were clear, it would be permitted in a Meshameres. If one could not drink it, it would be forbidden even in a scarf or a wicker basket, for in any case this is Borer.

Ran (DH Gemara): Most people are not particular [to strain] clear wine or water, therefore one may strain them even in a Meshameres, to make them extra clear. One could drink cloudy wine or water, but most people are particular [not to]. Therefore, one may strain them only through a Shinuy, i.e. through a scarf or a wicker basket, like the Mishnah says, but not in a Meshameres. Bein ha'Gitos one may mix [it and strain it], for most people drink it this way. Alternatively, Bein ha'Gitos the dregs did not yet separate from the wine, so all the wine is like one entity, so Borer does not apply. The Rambam says so, and it seems that the Rif agrees. If it is totally cloudy and one cannot drink it at all, one may not strain it even through a scarf or a wicker basket. I do not understand the Rambam in Perek 8 and 21.

Rambam (Hilchos Shabbos 8:14): One who strains wine, oil, water or other liquids in their Meshameres is liable if he strained k'Grogeres (the volume of a fig). One may strain wine that has no dregs or clear water in a scarf or a wicker basket, so it will be extra clear.

Rambam (21:17): Straining dregs is a Toladah of Borer or sifting. Therefore, even though one may strain clear wine or water in a scarf or a wicker basket, one may not make a cavity like he does during the week, lest he come to strain with a Meshameres.

Rosh (20:6): Bein ha'Gitos, the wine is still bubbling. It is not clear, and it is not cloudy. One may put it in a Meshameres.

Poskim

Shulchan Aruch (OC 319:10): One may strain clear wine or water in a Meshameres.

Mishnah Berurah (34): Clear is what most people would drink without straining. Some permit only if all would drink without straining, and he strains only to make it extra clear.

Rema: Even though is has fine Kismin (perhaps these are shriveled pieces of grape peels - PF), since one could drink it without straining, it is permitted.

Mishnah Berurah (35): Semag says 'a few Kismin', for if there are many, most people would not drink it like this.

Kaf ha'Chayim (81): The Bach and Eliyahu Rabah permit only if it is totally clear with only a few Kismin. The Acharonim hold like the Ran and Taz, who permit as long as most people are not particular.

Kaf ha'Chayim (84): One may blow straw or Kismin to the side, for one could drink without doing so.

Bi'ur Halachah (DH Ho'il): An Istnis (finicky person) who cannot drink it with Kismin or similar things, even if most people are not particular about this at all, we do not ignore his opinion and base the Halachah on normal people. For him, it is Borer!

Kovetz Teshuvos ha'Gaon R. Y.S. Elyashiv Ztz"l (1:37): If usually people are not particular, and drink water that did not pass through a strainer in the tap), Borer does not apply. However, it is forbidden for an Istnis.

Orchos Shabbos (1:3:39): He and ha'Gaon R. S. Vozner Shlita say similarly regarding a water filter. Ha'Gaon R. N. Karlitz Shlita permits, for one does not see the Pesoles (waste) in the water and not on the filter [until using it for a long time]. Ha'Gaon R. Shmuel Auerbach Shlita forbids, for even so this is Breirah. It is not clear whether one who is not particular may strain for an Istnis. The Bi'ur Halachah (302:1 DH v'Hu) was unsure whether one who is not particular about shaking dew off clothes may do so for another, who is particular.

Ashrei ha'Ish (2:56, footnote 97): The same applies to one who is particular to strain the water due to concern for bugs, e.g. in some places in the U.S.

Shulchan Aruch (ibid.): One may not strain water in a scarf, due to Libun (laundering).

Taz (7): The Beis Yosef (320) brought the Ran, who says that [also] wine launders! He brings a proof from Sukah 40a, which expounds "to eat, and not for Libun." This requires investigation. See what I wrote in 320:12.

Mishnah Berurah (36): This refers to a scarf not designated for straining. If it is, this is a Meshameres!

Shulchan Aruch (ibid.): Wine and other liquids are permitted.

Beis Yosef (DH v'Lich'orah): Ze'iri permits putting clear wine in a Meshameres, and a Beraisa permits mixing wine with its dregs and putting it in a Meshameres. It seems that this is a proper Meshameres made for this. If so, the Mishnah, which permits only in a scarf or a wicker basket, must discuss unclear wine. The Magid Mishneh says so in the name of the Rashba. However, if there are any dregs at all, he forbids even in a scarf or a wicker basket, for this is proper Borer. Rather, it is cloudy. Also the Ran says so. It seems that Rashi agrees. He says that the Mishnah discusses straining wine from Kemach (mold), for if it were clear, it would be permitted even in a Meshameres. If there were dregs, it would be forbidden even in a scarf or a wicker basket. The Tur holds like this, and therefore permits in a Meshameres clear wine or water, or wine with its dregs Bein ha'Gitos. However, it seems that he [always] permits in a scarf or a wicker basket even if it has dregs, for he rules like his father, and unlike the Rambam, who forbids. However, perhaps the Tur says about anything cloudy 'it has dregs', but if it has so much dregs that people do not drink it, also he forbids even in a scarf or a wicker basket, like the Meforshim. It seems that the Rambam decrees to forbid even clear wine or clear water in a Meshameres, for sometimes there is a Chiyuv Chatas for a Meshameres. He forbids unclear wine or water even in a scarf, due to a Meshameres. Ze'iri and R. Shimon ben Gamliel do not discuss a real Meshameres. Rather, they permit Leshamro (to strain). They call putting in a scarf or a wicker basket Meshameres. Ze'iri explains the Mishnah, that the Heter for a scarf is only for clear wine. Why did the Magid Mishneh overlook this?!

Rebuttal (Taz 9): This is difficult. Why should the Rambam say that Meshameres is not literal?! Rather, the Rambam explains that Ze'iri discusses what is so clear that it has no dregs, just Kemach or something else that fell in. Only this is permitted in a Meshameres.

Gra (DH v'Im, according to Damesek Eliezer): The Tur holds that the Mishnah mentioned a scarf to teach [through inference] that one may not strain water through it [but wine is permitted even in a Meshameres]. The Gemara discusses a real Meshameres. The Rambam holds that the Mishnah mentioned a scarf to teach that even it is permitted. All forbid a real Meshameres. The Gemara does not discuss a real Meshameres. Also wine in the Mishnah is not precise [to exclude water]. The Rambam holds that all squeezing is forbidden due to Libun. Therefore, Libun applies also to wine. The Ran proves this from Sukah 40a. We must say like Semag says, that because he does not intend to launder, we do not say that soaking is Libun. Therefore, even water is permitted.] The Tur holds that Libun does not apply to wine, but it applies even without intent. He said even in a scarf to teach unlike the Magid Mishneh and Ran, who hold that the Mishnah mentioned a scarf for the Chidush that it is permitted, i.e. even for cloudy wine, as long as there are no dregs and people can drink it. A Meshameres is permitted if it is clear.

Shulchan Aruch (ibid.): If the wine or water is cloudy, one may not strain it.

Taz (8): We forbid in a scarf if it is so cloudy that one cannot drink it. However, it may be so cloudy that most people would not drink it.

Mishnah Berurah (40): If people could drink wine or other liquids without straining, one may strain them in a scarf, for this is abnormal.

Shulchan Aruch (ibid.): The Rambam forbids even clear wine or clear water in a Meshameres. Even in a scarf he permits only what is clear, but not cloudy.

Mishnah Berurah (41): The Halachah follows the first opinion.

Mishnah Berurah (42): Even if they are a little cloudy, it is good to be concerned for the Rambam.

Kaf ha'Chayim (89): L'Chatchilah one should be concerned for the Rambam's opinion. When needed, one may rely on the first opinion, for it is primary.

Shulchan Aruch (ibid.): Bein ha'Gitos, as long as it is bubbling, one may mix the barrel with its dregs and put it in a scarf.

Beis Yosef (DH v'Yayin): Rashi explains that Bein ha'Gitos, all wine is cloudy, and people drink it with the dregs. Straining it is not Tikun, for one could drink it without straining. The Rambam permits Bein ha'Gitos as long as it is bubbling, for the dregs did not separate well from the wine, and all the wine is like one entity. The Ran says that the Rif connotes like this.

Mishnah Berurah (43): When all the wines are cloudy, people drink them with the dregs. Even if he mixes it up, it is as if it is clear.

Kaf ha'Chayim (90): This is according to the Rambam. The first opinion permits with Kismin, and all the more so in this case.