Question: May one clean a garment which became dirty with dust or brush a felt hat on Shabbat?
Answer: The Gemara (Shabbat 147a) states that one who shakes out a Tallit on Shabbat is liable a Chatat offering. This means that if one shakes out one’s garment on Shabbat in order to get out any dirt there may be on it, one is liable to bring a Chatat offering, for the shaking out of the garment is considered its laundering and laundering is a forbidden work on Shabbat.
The Rishonim nevertheless disagree about the interpretation of the above Gemara. According to Rashi, one who shakes out a garment to get out the dirt on it transgresses the prohibition of laundering on Shabbat. Thus, according to Rashi, it is forbidden to shake out a black garment which was soiled with dirt or dust because of the prohibition of laundering on Shabbat. The Ben Ish Hai (Parashat Vaychi) rules accordingly. This ruling is based on the words of the Rama (in his notation on Chapter 302) that it is preferable to be concerned with Rashi’s opinion.
Nevertheless, Maran Rabbeinu Ovadia Yosef zt”l writes that most Rishonim do not explain the above Gemara like Rashi does. Rather, they explain that “shaking out a Tallit” refers to when the Tallit or other garment becomes soiled from rain water that fell on them and the like in which case shaking out the garment will constitute the forbidden work of laundering on Shabbat. However, regarding anything which is completely dry, such as dust, there is absolutely no prohibition to shake out the garment. Indeed, Maran Ha’Shulchan Aruch rules likewise based on the words of Rabbeinu Chananel and several other great Rishonim who rule leniently even with regards to a new, black garment.
Thus, one may shake the dust out of a garment on Shabbat and this does not constitute the prohibition of laundering on Shabbat according to Maran Ha’Shulchan Aruch.
Although we cannot delve into all the fine details regarding this matter, we should nevertheless point out that even according to the opinion which prohibits this, the prohibition will only apply to a new, black garment which people are usually meticulous not to wear when it is dusty. However, if people do not care to wear it in this manner, there is no prohibition to shake out such a garment according to all opinions.
Regarding cleaning a garment or hat using a brush designated for this purpose, Hagaon Tiferet Yisrael writes that it is customary to act stringently regarding this manner because it is an action that appears to be done on weekdays. On the other hand, Maran Rabbeinu Ovadia Yosef zt”l once again writes that according to the letter of the law, there is room for leniency in this regard for our Sages only forbade actions usually performed only on weekdays when it is a matter that can eventually evolve into a Torah prohibition. However, cleaning a garment with a brush does not fall into this category. (Halichot Olam, ibid.)
Summary: According to Maran Ha’Shulchan Aruch, it is permissible to shake out or clean with a brush any garment which became soiled by dust and the like on Shabbat.