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- There are three methods that are considered laundering in Hilchos Shabbos: 
- Wetting a garment 
- scrubbing a garment and
- squeezing a wet garment.
- Another issue that arises is dusting off a garment. While the Gemara says that ruling only applies to new, black, garments whose cleanliness you care about, the Achronim point out that those rules are only a function of how concerned a person is about their clothing. Therefore, today when most people care about the appearance of their clothing, especially their Shabbos clothing, we must assume that it is prohibited to dust off the clothing.
- Many poskim hold that wet clothing that were left out to dry and you knew that normal circumstances it would be dry before the end of Shabbat you can use it when it dries.
In the Mishkan
- Melabain was performed in the mishkan in the preparation of the wool that would be used for the curtains. 
- After having washed one’s hands on Shabbat, it’s permissible to dry them on a towel, however, it’s preferable to rub one’s hands together to dry them before drying them on a towel. 
- If one’s hands became soiled with mud one may not clean them off by wiping them on a towel or handkerchief however one may wipe them on a rag which one isn’t concerned about cleaning. 
- One may dry dishes on a towel even if it gets wet but one may not dry out a thin cup or thermos. 
- If a drink spills on a tablecloth it is permissible to place paper napkins over the wet area to dry up some liquid. However, one may not press it against the wet area to draw out the liquid. 
- One may wet a paper towel or disposable napkin in order to wipe down a sticky counter-top or plastic tablecloth. However, one may not squeeze or press the wet paper towel or napkin. 
- There is a prohibition against wetting a garment lest you come to squeeze it out (which is an isur deoraisa). This prohibition is aside from the Torah prohibition of wetting a garment. Therefore even if one finds himself in a situation in which there is no Torah prohibition to wet a garment (like derech lichluch) he must take care that he does not wet the garment to the extent that he is likely to squeeze it out by accident.
- The 39 Melachos (p. 693)
- Wetting a non-leather garment is Melabain according to the Mishna on Shabbos 142. There are exceptions to this rule as there are various Gemaras that allow one to wet clothes on Shabbos. The Rishonim offer several explanations: Tosfot Shabbos 111 s.v. Hai writes that wetting is only cleaning if the garment was previously dirty but it seems like the Rashbam argues in Tosfot 120 s.v. noten. Tosfos Shabbos 111 s.v. hai also suggests that wetting is not considered cleaning if it is done in a way that soils the garment. Hagahot Maimoniyot Shabbos 22:80 says that wetting is only considered cleaning if the individual had intent to clean.
- Gemara Shabbos 147. Tosfos s.v Hana’er Tallito says that this is only an issue when it comes to wet dirt, while Rashi there argues that it even applies to dry dirt.
- Aruch Hashulchan 302:3, Biur Halacha 302:1 s.v. aleha
- The 39 Melachos (p. 709). Rabbi Ribiat quotes Biur Halacha 302:1 s.v. lechush that one may be lenient by light colored garments and brush off the dust with a shinui
- Orchot Shabbat v. 2 p. 609-11 explains that Minchat Yitzchak 1:81, Shevet Halevi 1:62, Igrot Moshe 5:22:26, and Minchat Shlomo 1:10 all hold that as long as the wet clothing will dry on their own it isn't muktzeh after they dry. Rav Nevinsal in Byitzchak Yikareh on m”b 308:63 agrees. Orchot Shabbat cites an oral communication in which the Chazon Ish agreed. Dirshu 308:89 cites this. Shvut Yitzchak ch. 8 citing R' Elyashiv argued that gamru byadey adam it isn't applicable to muktzeh machmat isur. Orchot Shabbat asks from Tosfot Shabbat 43a. Rav Moshe in Igrot Moshe 5:22:26 (Tiltulei Shabbat p. 250) adds a caveat that it would have to be you took out the clothing to dry but not that you left it in the washing machine.
- Rashi Shabbos 73a s.v. Hemelabeno says that the Av Melocho of Melabain that took place in the mishkan was the removing the wool of its impurities. The Rambam Hilchos Shabbos 9:11 argues that the Av Melocho was the bleaching of the wool. The Rambam still agrees that cleaning wool is prohibited since it falls under the Toldah of Mechabeis, laundering.
- Rambam Mishneh Torah Hilchot Shabbat 9:10
- S”A 302:10 writes that after having washed one’s hands, it’s preferable to rub one’s hands together before drying them on a towel. The Rama 302:10 writes that the minhag is to be lenient not to be concerned for this issue. The Mishna Brurah 302:50 explains that this is based on the Rabbeinu Tam quoted by Tosafot Shabbat 111b who says that it is permissible because even though one is getting the towel wet since it is effectively dirtying the towel there is no issue of Mechabes (cleaning). Nonetheless, because there are some authorities who are strict it’s a good practice to be strict. The Yalkut Yosef (Shabbat vol 2 pg 89) agrees that it’s permissible but preferable to follow Shulchan Aruch.
- S”A 302:11 writes that if one's hands became soiled with mud one may wipe them on a hard cloth used to hold thorns but not a regular hand towel because one may come to clean the towel. Mishna Brurah 302:57 writes that a rag which one is not concerned about is just like a hard cloth and one may wipe one's hand on it. This is also the opinion of Shemirat Shabbat KeHilchata 14:26 (in new editions, 14:29) seems to add that a paper towel is like a rag that one isn't concerned about.
- S”A 302:12, Mishna Brurah 302:59, Shemirat Shabbat KeHilchata 12:19, 21
- 39 Melachos (Rabbi Ribiat, vol 3, pg 700) writes if a drink spills on a tablecloth it is permissible to place paper napkins over the wet area to dry up some liquid. However, one may not press it against the wet area to draw out the liquid. Shemirat Shabbat KeHilchata 12:38 (in new editions 12:41) and Shabbos Home (pg 183) agree.
- Rav Moshe Feinstein in Iggerot Moshe 2:70 rules that one may wet a paper towel in order to clean the table because one is not concerned to squeeze out the liquid and also the paper towel gets ruined by the water. 39 Melachos (Rabbi Ribiat, vol 3, pg 694-5) agrees but adds that it's preferable to sprinkle water on the sticky surface and then wipe it with a dry napkin. See Shemirat Shabbat KeHilchata 12:40 (in new editions 12:43) who says that one should wipe down the dry table with a dry cloth and does not suggest using a wet paper towel.
- Mishna Brurah 302:46. There are examples of cases where you need not worry, for example where you are not bothered if water is absorbed in such a garment, or if there is not too much water.