Squeezing and drying hair on Shabbos:
Biblically, there is no prohibition against squeezing hair on Shabbos. Nevertheless, it is Rabbinically forbidden to squeeze liquid out from hair on Shabbos [due to the Mifarek prohibition]. Thus, every bather needs to beware not to squeeze his hair.
Q&A on squeezing and drying hair on Shabbos
What areas of hair contain the squeezing prohibition?
Only those areas which have hair growing in close proximity to each other such as the hair of the head, beard [underarms and pubic area]. However, the hair of the arms [and legs] are distanced from each other and thus do not contain a squeezing prohibition.
May one squeeze the water out from his eyelids after leaving the Mikveh?
This matter is dependent on the dispute mentioned above. Practically, one may be lenient to do so as otherwise there is pain involved in having the water get into one’s eyes, as well as that since the hair of the eyelids are not close together they most probably do not contain a squeezing prohibition at all.
If while drinking one wet his mustache, may he squeeze the water out?
One is to be careful to only do so using a towel as opposed to one’s hands.
May one scratch his hair when it is wet?
Yes. Although doing so will inevitably cause water to get squeezed out it is nevertheless permitted to be done. Nevertheless, it is best to do so using one’s hat or the like.
May one wash one’s beard/head and the like on Shabbos?
The law: Some Poskim rule it is forbidden to wash the hair on Shabbos using one’s hands, due to that in doing so one causes the water to be squeezed from his hair. Other Poskim, however, rule it is permitted to wash one’s hair on Shabbos, even if one will squeeze the water in the process of washing. Practically, although doing so in it of itself is not prohibited from the letter of the law, the custom is to avoid doing so being that one may come to squeeze his hair in a way that is prohibited.
In case of Discomfort: In a case of discomfort one may wash his head.
In case of dirt: If one’s head or beard contains dirt it may be washed.
Washing ones bearded face when no intent to wash the beard: As well one may always wash and dry his face despite the fact that water will inevitably get in his beard and be squeezed, as the Sages did not uphold their decree in such a case.
May one use soap: From the letter of the law, some write that it is permitted to even use liquid soap to wash the hair, so long as one is careful not to squeeze the hair in the process. Nevertheless, practically one is not to do so, and so is the custom.
Q&A on using a towel
May one dry his head or beard or other hairy area with a towel?
Yes. Nevertheless, one is to only do so gently, without force. However, there are those who are scrupulous not to dry the hair of the beard and rather shake out the water and let it dry on its own.
May one leave a towel over his wet hair letting the water get absorbed into the towel on its own?
Yes, doing so is permitted even according to the stringent opinions mentioned above.
 Admur 326:6; Raba and Rav Yosef in Shabbos 128b
The reason: As hair is hard and does not actually absorb water inside of it. [Admur ibid; Rashi ibid] The water is absorbed between each individual hair. It hence does not carry a squeezing prohibition of Mifareik of laundering which is relevant to clothing and other items that are squeezed.
 Admur 326:6 and in 330:1; M”A 326:8 and 320:23; Beis Yosef 330; Kesef Mishneh 2:11; Implication of Rambam 9:11; Kol Bo and Maharil 139 brought in M”A ibid and Beis Yosef Y.D. 199; M”B ibid and 326:24, 25, 320:55; Biur Halacha 302 “Assur”; Aruch Hashulchan 320:35; Kaf Hachaim 320:110
Other opinions: Some Poskim rule there is no prohibition involved in squeezing hair. [Implication of Gemara ibid and Meiri Nidah 67b; Rashi and Rabbeinu Tam according to Avnei Nezer 157:14; Chida in Shiyurei Bracha 199:3 based on Beis Yosef Y.D. ibid; See Imros Moshe p. 332 for a lengthy discussion on the opinions on this matter.]
Does the laundering prohibition apply to hair? No. [Avnei Nezer ibid]
 Admur 326:6; M”A 326:8; M”B 326:21
 Ketzos Hashulchan 133 footnote 8; Piskeiy Teshuvah 326:7; Shabbos Kehalacha 18:53
 Ketzos Hashulchan ibid; Minchas Shabbos 86:6; Rav SZ”A in Maor Hashabbos 4:44; Shabbos Kehalacha ibid
 Yeshuos Chochmah brought in Ketzos Hashulchan ibid; Az Nidbaru 13:10
 Yeshuos Chochmah brought in Minchas Shabbos ibid; Az Nidbaru 13:10; Rav SZ”A in Maor Hashabbos 4:44;
 Piskeiy Teshuvos 326 footnote 133
 Ketzos Hashulchan 133 footnote 8
 And in a case of pain the Sages did not uphold their decree.
 Ben Ish Chaiy Pekudei 2:8; Ketzos Hashulchan 133 footnote 8
 Minchas Shabbos 86:6
 The reason: As a) One does not desire the water to get squeezed and it is thus a Pesik Reishei which is not Nicha Lei. As well b) In a case of pain the Sages did not apply their decree.
 Avnei Nezer 157:14; Sheivet Halevy 5:45; Mishneh Halachos 5:51; They are stringent to avoid washing the beard individually with ones hands [as opposed to when using a shower head or entering into a Mikveh] due to that it will inevitably cause hair to get squeezed out.
 M”B 326:25; Implication of Michaber 326:9 and Admur 326:8; Ashel Avraham Butchach 320:17; Ketzos Hashulchan ibid; Igros Moshe 1:133; Piskeiy Teshuvos 326:11; Rav SZ”A in his final ruling in SSH”K 14 footnote 1; See Shabbos Kehalach 18 footnote 70; Implication of Ben Ish Chaiy/Minchas Shabbos/Ketzos Hashulchan, as they allow to dry the hair. Furthermore, from the fact that Admur never mentions not to wash the beard or head when mentioning the ruling that one may wash minority of the body itself proves that according to Admur washing these areas contain no prohibition and it is only their squeezing that is problematic. Vetzaruch Iyun from SSH”K 14:11 which plainly rules that one is not to wash these areas, thus contradicting the ruling that he brought earlier in footnote 1 in the name of Rav SZ”A.
 The reason: As in addition to the reason he mentions there [that the water gets squeezed towards the body] the water squeezed goes to waste. Likewise, the Sages did not want to decree against one washing his face on Shabbos. [See Ketzos Hashulchan ibid]
 Ashel Avraham Butchach 320:17; Sheivet Halevy 5:45; Mishneh Halachos 5:51; Piskeiy Teshuvos 326:11; See Ketzos Hashulchan 133 footnote 8
 Piskeiy Teshuvos ibid
 Piskeiy Teshuvos ibid ; Igros Moshe 1:133; Peri Megadim M”Z 320:12. However there are opinions [Sheivet Halevy 5:45; Mishneh Halachos 5:51] who are stringent in this.
 Rav SZ”A 14 footnote 9; Piskeiy Teshuvos 326 footnote 132; Shabbos Kehalacha 18:54
 The reason: As although water is squeezed out. Nonetheless other water comes right back in. [ibid]
 Ketzos Hashulchan 133 footnote 8
 Piskeiy Teshuvos 326:11 based on Michaber 326:9 and Admur 326:8 and M”B 326:25 and 28 which allows using Morson on the beard
 Sheveit Halevi 5:45; Piskeiy Teshuvos ibid
 See Shabbos Kehalacha 18:57; Piskeiy Teshuvos 326:12
 Ben Ish Chaiy Pekudei 2:8; Minchas Shabbos 8:6; Ashel Avraham Butchach 320:17; Ketzos Hashulchan 133 footnote 8; Kaf Hachaim 320:111; Shabbos Kehalacha ibid
Other opinions: Some Poskim prohibit using a towel to dry one’s hair even lightly. [Az Nidbaru 1:55; Beir Moshe 1:31] Practically, one may rely on the great Poskim listed above who permit doing so, especially being the entire prohibition is of Rabbinical nature, and one has no intent to squeeze the hair, and Admur never mentions in the laws of drying the body [in 301 and 302] any need to be careful in the above.
The reason: Now although one will inevitably squeeze out some water in the process, nevertheless it is allowed as the squeezing is done in an irregular fashion [using the towel], as well as that the water squeezed goes to waste. As well as regarding the beard, since one who does not dry his face it can break out in a skin problem, it was never decreed against drying it, even though it is inevitable that one will not squeeze out some of the water from his beard in the process. [Ketzos Hashulchan Ibid] Likewise it is permitted because the squeezed liquid is not recognizable and enters straight into the towel. [See Shabbos Kehalacha ibid footnote 133]
 Ben Ish Chaiy ibid; Kaf Hachaim 320:111; Ketzos Hashulchan ibid; Shabbos Kehalacha ibid
 Ben Ish Chaiy ibid; Keztos Hashulchan ibid
 Az Nidbaru ibid; Piskeiy Teshuvos ibid; Shabbos Kehalacha ibid