Gemara

(Rav Huna): If one shook out his Talis on Shabbos, he is Chayav Chatas.

This is only if it is new. If it is old, it is permitted;

This is only if it is black. If it is white or red, it is permitted;

This is only if he is insistent. If he is not insistent, it is permitted.

Rav Yosef asked Abaye to bring his hat; Abaye saw dew on it, and was hesitant.

Rav Yosef: Shake it out. We are not insistent!

Rishonim

Rambam (Hilchos Shabbos 10:18): If one shook out a new black Talis on Shabbos to beautify it and remove the Tzihuv Lavan (yellowing part; some texts - white wool) from it, like the craftsmen do, he is Chayav Chatas. If he is not insistent, it is permitted.

Ra'avad: Other Meforshim explain that he shakes it out from water that fell on it. This is a Toladah of laundering. This Perush is like the context. (The Gemara before and after this discusses leaving to dry or shaking out garments that became wet.)

Rosh (22:10): Rashi explains that he shakes out his Talis from the dust on it. R. Chananel explains that he shakes out the dew. This is laundering. The Gemara connotes like this, for right after this, it discusses shaking out a hat from dew. There is no source that laundering applies to shaking off dust. Rashi explains that 'he is particular about it' means that he would not wear it without shaking it out.

Rashba (147a DH ha'Mena'er): For one who is particular, it is a Pesik Reishei. For one who is not particular, it is Davar she'Eino Miskaven, for it is not total laundering.

Poskim

Shulchan Aruch (OC 302:1): One who shakes out a new black Talis from the dew on it is liable, for shaking is good for it like laundering.

Beis Yosef (DH ha'Mena'er):. Also Tosfos, the Ran, Mordechai, Ra'avad and Magid Mishneh in the name of the Rashba explain like the Rosh and R. Chananel, that he shakes out the dew.

Beis Yosef (DH u'Mah she'Chasav v'Davka): Rashi explains that dust mars the appearance of black garments. R. Chananel must say that shaking out the dew from black garments beautifies them, and it is recognized that they were laundered.

Mishnah Berurah (1): 'New' is as long as he has not used it much, and it still looks new.

Kaf ha'Chayim (1): It does not depend on whether or not he laundered it yet. It depends only on whether he used it much.

Mishnah Berurah (2): If he is not particular, and sometimes wears it without shaking it out, he may shake it out l'Chatchilah.

Bi'ur Halachah (DH v'Hu): If one is particular, this is improvement of the garment. If Reuven shakes for Shimon, and Reuven is particular but Shimon is not, one should be stringent, for it is improvement for the shaker, and he does the Melachah. However, since there is no improvement for the owner, perhaps this is not considered Melachah. Perhaps this was the case of Rav Yosef's hat. Abaye was hesitant, for he was particular, and Rav Yosef answered that it depends on the owner. Had it been oppositely, it would have been forbidden. This requires investigation.

Kaf ha'Chayim (3): Similarly, one may not shake out snow.

Bi'ur Halachah (DH Min): Eliyahu Rabah brings from Agudah that the same applies to snow. I say that if the snow did not yet melt at all, it is permitted, like the Rema (below) permits feathers, for they merely stand on top of the garment, but do not enter it. However, he must shake gently. Even if it melted slightly and was not recognized, only the [solid] snow on top will come off.

Shulchan Aruch (ibid.): This is if he is particular not to wear it without shaking it out.

Bach (1): Rashi derived that regarding Rav Yosef, the Gemara mentioned that there was dew on the hat. Here, it mentioned Stam shaking out. This shows that even shaking out from earth is laundering. Or Zaru'a said that one should be concerned for Rashi's opinion. Semag says that R. Chananel explains like Rashi. Also the Mordechai rules like Rashi; his words are from Sefer ha'Terumah. However, he says that we are not particular. L'Chatchilah one must be careful not to shake out from earth or water.

Bigdei Yesha (on Mordechai, 9): In Remez 441, the Mordechai explains like R. Chananel and says that Rashi's Perush is difficult. Here (Remez 444) he says "we conclude that one who shakes out a Talis from earth is Chayav Chatas, if it is new, black and he is particular. We are not particular about wearing it with earth or dew, so it is permitted." It seems that here he does not come to teach what he shakes out, for he already explained this. Rather, he comes to teach that all agree that if he is not particular, it is permitted. This is obvious according to R. Chananel, for Rav Yosef explicitly said so regarding dew! He mentioned Rashi, for it was not obvious regarding earth.

Note: I do not understand why it was clear regarding dew. Rav Yosef said 'we are not particular.' Perhaps he refers only to his city! Perhaps people's mindsets changed from the time of the Gemara until now!

Kaf ha'Chayim (9): The Tur, Shulchan Aruch and Acharonim brought this law. This shows that we do not hold like the Mordechai, who says that nowadays we are not particular.

Rema: All the more so, one may not shake out a garment that fell in water or on which rain fell. This refers only to a new garment that one is particular about it.

Magen Avraham (2): The Kolbo says that one is particular about a new garment, for it shrinks in water. This implies that Stam, one is particular about a new garment. Even though regarding dew the Gemara said '[even if it is new,] this is only if he is particular about it', water is different, for it makes garments shrink. Sefer ha'Zichronos forbids all garments.

Kaf ha'Chayim (5): He means that Sefer ha'Zichronos forbids garments of all colors. Levushei Serad says that the Acharonim concluded like this, for Kolbo says that the concern is squeezing, and this applies to all colors.

Gra (DH v'Davka): The Gemara said that it is a problem only for black garments, i.e. regarding dew or a small amount of water. Then one is particular only for black garments. The Isur is due to squeezing. It is only if one is particular. Regarding much water, surely he is particular. Ri received from R. Tam that it is only if he shakes his Talis due to water or rain that fell on it.

Mishnah Berurah (4): Even if it is old, one must be careful not to shake out the water vigorously, lest he squeeze. When a garment is soaked, shaking it out squeezes out the water well.

Rema (ibid.): Some forbid shaking out a garment from the dust on it, if he is particular about it. It is good to be concerned for this opinion.

Magen Avraham (3): This opinion forbids only if the garment is new and black.

Mishnah Berurah (5): The Rema said 'forbidden', even though Rashi says that one is liable, to teach that if he is not particular, it is permitted l'Chatchilah.

Mishnah Berurah (6): Eliyahu Rabah says that we should forbid according to letter of the law, for many Rishonim hold like Rashi. In any case one may rely on the first opinion to be lenient through a Nochri, especially since sometimes this pertains to Kevod ha'Briyos. When one goes to the Beis Midrash, it is good to put one's hat and garments in a safe place, lest they fall to the ground and get dusty, and he will come to Chilul Shabbos.

Kaf ha'Chayim (7): Even when one may shake it out, he must do so gently, for if not, he will squeeze.

Bi'ur Halachah (DH Lachush): The Chayei Adam says not to remove it by hand, and all the more so through a brush or Matelka (rag?). He brings from Ya'aros Devash that we are not experts about what is considered new, therefore one must be careful about all garments. The other Acharonim did not mention this. One who is stringent will be blessed, and one who is lenient has whom to rely on. It is good to do so through a Shinuy, i.e. awkwardly. L'Chatchilah one should not come to such a situation.

Kaf ha'Chayim (1): Also Minchas Pitim, Orchos Chayim and Ben Ish Chai are stringent like Ya'aros Devash.

Rema (ibid.): One may remove feathers from a garment on Shabbos.

Magen Avraham (4): Sefer ha'Zichronos (53) is concerned for the Isur of tearing. Therefore, if one's clothes because tangled with thorns, he should separate them covertly and slowly, lest they tear. If they tore, he is exempt, for he did not intend.

Gra (DH Aval): The Yerushalmi permits one whose clothes because tangled with thorns to separate them. All the more so here it is permitted. Rashi explained that the dust ruins the appearance, but not of white garments. All the more so [one may remove] feathers!

Mishnah Berurah (7): This is not like laundering, for the feathers are not inserted in it like water or earth. They merely stand on top of the garment. Therefore it is permitted even for a new garment, even if he is particular.

Kaf ha'Chayim (8): Pesach ha'Dvir says that if one leaned on a wall and the plaster stuck to his garment, he may snap the middle finger on it to remove it. However, if thick moist plaster stuck to it and now dried, one must be stringent.

Rema: See below 337:2.

Bi'ur Halachah (DH v'Ayen): There it says that even when one may remove dust (e.g. an old garment, or he is not particular) or feathers, one may not use a broom from fine chips, lest they break. Nowadays we permit only by hand (Tif'eres Yisrael, Ya'aros Devash).