Men dyeing hair:[1]
It is [Biblically[2]] forbidden for a man to dye white hair to a black color [or other beautifying color[3]].[4] [It is likewise forbidden to dye black or other colored hair to a different beautifying color.[5]] This prohibition applies to even a single hair.[6] [This prohibition applies even to a temporary dye.[7] This prohibition applies even in a time of need, even if not dyeing the hair will make one look shameful and embarrassing.[8] Some Poskim rule that in areas that men are accustomed to dye their hair then it is permitted for men to dye it, as explained in Q&A!]

Black to white-a degrading color:[9] It is permitted even initially for even men to dye a black hair to a degrading color, such as a white color.[10]

Women: It is permitted for a woman to dye their hair without any restriction.

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[1] Michaber 182/6; Tur 182; Rambam Avodas Kochavim; Ohel Yaakov p. 382-391] short, with details is long

[2] Tur 182 in name of Rambam that one who does so is liable

[3] Shevet Hakehasi 2/257-2

[4] Michaber ibid; Tur ibid; Rambam ibid

The reason: As doing so is a beautification that is done by women and is forbidden due to the prohibition of Beged Isha. [Taz 182/7; Biur Hagr”a; Beis Yosef 182] It is done in order so one look young and handsome, hence falling under the above prohibition. [Beis Yosef ibid]

[5] Chelkas Yaakov 87; Mishnas Yosef 8/17; Shevet Hakehasi 2/257

[6] Michaber ibid; Tur ibid; Rambam ibid

[7] Shevet Hakehasi 2/257-2; Poskim brought in Ohel Yaakov 79 footnote 107

[8] Darkei Teshuvah 182/17 in name of Poskim

[9] Taz 182/7; Beis Yosef 182 in correct Girsa of Rambam ibid; See Mishnas Yosef 8/17 regarding if the dye does not make the person look more handsome.

Other opinions: The Tur writes in name of the Rambam that one who dyes a black hair white is liable. The Beis Yosef and Taz ibid attribute this to a misprint, as in truth there is no prohibition involved in such a case.

[10] The reason: As doing so is not a beautification that is done by women, as it is done to make one look old and respectable. [Beis Yosef ibid]

Other opinions: The Tur writes in name of the Rambam that one who dyes a black hair white is liable. The Beis Yosef and Taz ibid attribute this to a misprint, as in truth there is no prohibition involved in such a case.

[11] See Igros Kodesh 6/268, [printed in Shulchan Menachem 4/133]; Ohel Yaakov 81

Background: There are three levels in Beged Isha: a) A matter which is Biblically considered Beged Isha, such as armor. B) Matters which their inherent purpose is for beauty for a woman. C) Matters which are simply accustomed only by women. Level a) can never change irrelevant of the custom, level c) is dependent on the custom, however level b) can be viewed both ways. Dyeing hair is considered like b) and hence is subject to argument in whether its law changes upon the change of custom. [Igros Kodesh ibid]

[12] Imrei Yaakov 2/18 [Rav Yaakov Shtern]; Mesores Moshe of Igros Moshe p. 248; Leaning opinion of Rebbe in Igros Kodesh ibid however he does not conclude this way; See Ohel Yaakov 79 footnote 109

[13] The reason: As some of the prohibitions of Beged Isha only apply when men are not accustomed to perform the actions. [Michaber 182/1 regarding shaving] This possibly applies even if it is only common for gentiles to do so. [Perisha 182, brought in Rav Akiva Eiger 182; Igros Kodesh ibid] Now, the dyeing of hair falls under category b) brought above which seemingly is subject to change based on custom. [Igros Kodesh ibid]

[14] Setimas Haposkim; Shevet Halevi 3/2

[15] The reason: As dyeing hair is an inherent practice of women for purposes of beauty and can never be changed irrelevant of custom.

[16] See: http://articles.latimes.com/2012/jan/29/image/la-ig-mens-hair-color-20120129

[17] Igros Kodesh 8/155, [printed in Shulchan Menachem 4/136]; Igros Kodesh 6/268, [printed in Shulchan Menachem 4/133] the Rebbe leans to be lenient although than concludes that all above is “Lehalacha but not Lemaaseh in any way, and Bepoel the Achronim already gave their ruling”

[18] Igros Kodesh 8/155, [printed in Shulchan Menachem 4/135]; Betzail Hachochmah 5/127; See Chakira in Igros Moshe Y.D. 1/82; See Ohel Yaakov 79 footnote 113

[19] The reason: As perhaps the main prohibition is the result and not the action [See Igros Moshe ibid; Maharsham 2/243] As we find regarding Sirus that it is forbidden to consume chemical sterilization, and hence the same should apply here regarding dyeing. [See Igros Kodesh ibid]

[20] Meaning, in cases of great need when a Heter may anyways be given, such as for a Bochur who has white hairs to dye it black for Shidduchim, there is greater leniency if he does so through a pill than through doing so directly.