Removing dust/dirt from clothing:
*This Halacha does not apply to removing wet or dry mud from clothing-See Halacha ?? for further details on this matter!
Some opinions rule it is forbidden to shake off earth or dust from a new black colored clothing, if one is particular not to wear the clothing while dirty. [This prohibition applies even against shaking it off with one’s hand, and certainly against doing so with a brush. Other Poskim however rule there is no prohibition in shaking off the dust.] Practically, it is proper to suspect for their opinion [and so is the custom of all Jewry, both Sefaradim and Ashkenazim]. [According to all opinions however, one may shake dirt or dust off old clothing, or off non-black colored clothing, or off even new black clothing which one is not particular to avoid wearing while dirty. Accordingly, all white, red or other non-black colored clothing may have dust shaken off from them. Clothing that have had their black color fade due to the sun are possibly no longer considered black.]
Definition of new clothing: All clothing are considered new until they have been worn so many times that they no longer retain their new state of appearance. [This applies even if they have been laundered many times, if they nevertheless still appear new. Based on this, some Poskim rule one is to be stringent by all black clothing, as it is unclear when they become defined as having lost their new state. Clothing that have had their black color fade due to the sun are possibly no longer considered new.]
May one use a brush to brush the dust off permitted clothing: Those clothing that are allowed to have their dust shaken off using one’s hands, from the letter of the law, are also permitted to have one use a brush to [lightly] brush the dust off, so long as he does not use a brush which its strands will inevitably break in the process of the brushing. [Nevertheless, the custom is to forbid using any type of brush on clothing, even if the brush is made of hairs and it will not break in the process.]
How may one remove the dust off the permitted clothing?
May one brush dust/dirt off his black pants/Kapata?
May one brush dust off his black hat?
May one brush dust off shoes?
 Admur 302/3; Rama 302/1; M”A 302/3; Olas Shabbos 302/4; Ketzos Hashulchan 116/3
 Rav Huna Shabbos 147 according to Rashi; Shivlei Haleket 109; Sefer Hateruma 254; Or Zarua Shabbos 81; Semag LSL 65 in name of Rabbeinu Chananel
 The reason: This is forbidden due it being similar to laundering as the dust ruins the look of the garment, and this removal of the dust [beautifies the clothing and] is laundering it. [Admur ibid; Rashi ibid]
 Chayeh Adam; Ketzos Hashulchan 116 footnote 9
 Michaber 302/1; Tosafus Shabbos 147a in interpretation of Rav Huna Shabbos 147; Rosh; Ran
 The reason: As this is not considered the way of laundering. [Tosafus ibid]
 Ben Ish Chaiy Vayechi 2/8; Kaf Hachaim 302/8; Or Letziyon 2/24-1
 Admur ibid; Rama ibid; Bach 302; Or Zarua
 Admur 302/1 regarding shaking off dew and the same applies regarding shaking off dust; Admur 337/3 regarding using a brush to brush clothing
 Admur 302/1 regarding shaking off dew and the same applies regarding shaking off dust
 Ketzos Hashulchan 116 footnote 9
 Admur 302/1 regarding shaking off rain, and the same applies here as well; M”B 302/?; Ketzos Hashulchan 116 footnote 9
 Bach; Levush; Peri Megadim brought in Ketzos Hashulchan 116 footnote 2
 Yearos Devash, brought in Chayeh Adam, brought in Biur Halacha 302/1 “Lachush” and Ketzos Hashulchan116 footnote 2; Biur Halacha ibid concludes that the other Achronim made no mention of this ruling and hence, although one who is stringent is blessed, one who is lenient has upon what to rely; See Piskeiy Teshuvos 302/2
 Ketzos Hashulchan 116 footnote 9
 Admur 337/3 and Rama 337/2 regarding brushing clothing; Biur Halacha 302/1 “Yeish Omrim” that one may use a brush to remove the dirt
 See Biur Halacha 302/1 “Yeish Omrim” that one may not rub a stain off; See Piskeiy Teshuvos 302/2 footnote 28 who learns from here that it is forbidden according to all to forcibly use a brush to remove dust; See Q&A! Vetzaruch Iyun from wording of Biur Halacha ibid
 Tiferes Yisrael in Kalkeles Shabbos 13; Yearos Devash; Makor Chaim 302; Ketzos Hashulchan 146 footnote 62; Piskeiy Teshuvos 302/2; See however Biur Halacha ibid from whom it is evident that he did not hold of this prohibition
The reason: As it appears like fixing a vessel and is considered a mundane act. [Poskim ibid]
 Biur Halacha 302/1 “Yeish Omrim” based on Michaber 302/5 and Admur 302/12 who forbid rubbing a laundered shirt due to the whitening prohibition; Piskeiy Teshuvos 302/2
 Aruch Hashulchan 327/4; Shut Rav Chaim Zonenfeld 38; Shevet Halevi 5/37; Ateres Moshe 1/108; Bris Olam Miabed 3; Piskeiy Teshuvos 302/3 footnote 43 in name of Poskim and 327/5
 The reason: The definition of laundering by leather is different than regular clothing-see Admur 302/19!
 Shevet Halevi ibid; Ateres Moshe ibid; Bris Olam ibid; Piskeiy Teshuvos ibid