Must a married [or once married] woman cover every strand of hair of her head?
It is [Biblically, and at the very least Rabbinically] forbidden for a married woman to reveal even a single hair of her head to the public, and hence all of the hair, must be covered. It is for this reason that many of today’s Poskim rule one is to wear a wig [Sheitel] rather than a head Kerchief [Tichel], as the wig guarantees the covering of all the hair, as opposed to a head kerchief. However, there is a minority opinion amongst today’s Poskim who novelized that according to Halacha, a woman is permitted to be lenient to uncover up to two finger-worth’s [4 cm] of the top part of her hair. The Rebbe, as well as the vast majority of today’s Poskim, negate this opinion, and state it has no place in Shulchan Aruch, and so is the explicit ruling of the Tzemach Tzedek in numerous responses that every strand of hair must be covered. Those women who uncover even more than 4 cm of their hair are to be protested according to all. Furthermore, everyone agrees that according to the Zohar it is a complete prohibition to reveal any hair of the head, and one who is not careful in this causes poverty and spiritual challenges for their children, as well as other Tzaros r”l.
It is forbidden for a married, or once married, woman to reveal even one strand of hair from her head to the public, and hence a married woman must choose a hair covering that facilitates the covering of all of the hair of her head.
 See in great length Kevuda Bas Melech [Weiner] 1:2; Tefila Lemoshe [Levi] Vol. 5:50-54
 Dovev Meisharim 1:124; Chazon Ish in Dinim Vehanhagos 2:88-9 and in Teshuvos Vehanhagos 1:62
 See Dovev Meisharim 1:124
 Tzemach Tzedek Even Haezer 139:2, Y.D. 93:10, Shaar Hamiluim 1:45, and Mishnayos Brachos 3/5-3 from the fact he prohibits the sideburns; Chasam Sofer 36 “Any hair in any area of the head is an Erva”; See Mahariy Levi 9 who allows revealing some of the hair until the day after the Chuppah; Mayim Rabim on Tur 7, brought in Tzemach Tzedek Mishnayos ibid; Ashel Avraham Buchach “Proper to beware against even one hair”; Sdei Chemed; Teshuras Shaiy 1:51; Shulchan Haezer 9:10-5; Shulchan Shlomo 75; Vayeishev Yosef Y.D. 2; Dovev Meisharim 1:124 “G-d forbid to be lenient even regarding a small amount of hair, and it is Biblically forbidden”; Menuchas Moshe; Teshuvos Vehanhagos 1:62; Beis Baruch on Chayeh Adam 4:10; Lehoros Nasan 5; Divreiy Yisrael O.C. 34; Piskeiy Teshuvos 75:10; Chazon Ish in Dinim Vehanhagos 2:88-9 and in Teshuvos Vehanhagos 1:62; Shevet Halevi 5:15; Rebbe Ben Tziyon Aba Shaul brought in Kuntrus Kisui Rosh Leisha; Rav Mashash in Tevuos Shemesh 137; Shemesh Umagen 2:15-17; Rav Yaakov Yosef, as heard in his Shiur; Az Nidbaru 12:41 that one should cover every hair; Rav Moshe Shternbuch in Daas Vehalacha 1; Rav Lefkovich in Kovetz Beis Hillel 9; Or Yitzchak 3; Nitei Gavriel in Kovetz Or Yisrael 36-37; Zohar Parshas Nasso p. 239, brought in M”A 75:4; Maharam Alshiker 35; Chasam Sofer ibid; M”B 75:14; See Sefer Kevuda Bas Melech [Weiner] 1:2 footnote 8 and 10 for a lengthy discussion on this subject and the sources in Talmud and Poskim for the prohibition
 Poskim ibid including: Chazon Ish ibid; Rebbe Ben Tziyon Aba Shaul ibid; Or Yitzchak 3 in name of Rav Moshe Feinstein; Letters of Rebbe printed in Shulchan Menachem 6:80-84
 Igros Moshe 4:112; Even Haezer 1:58 [However, see Or Yitzchak 3 in name of Rav Moshe Feinstein that even he agreed that initially one should wear a wig in order to cover every hair]; Rav Ovadia Yosef as quoted in Yalkut Yosef Otzer Dinim to be lenient Bedieved, and so writes Tefila Lemoshe ibid 5:50-5 in his name in length that so he is accustomed to rule. [Rav Yaakov Yosef, however, stated that the only Rav who was ever lenient in this matter was the Igros Moshe. In other Shiurim, however, he stated that Syrian women [Chalabi] were accustomed to be lenient in this, however, women who came from other Sephardic countries [Yemen, Egypt, Morocco, Iraq] were very strict.] The Teshuras Shaiy ibid in a response initially tried to also learn this way in Maharam Alshiker, although he concluded stringently. See Sefer Tefila Lemoshe ibid for a lengthy discussion on this subject, and he concludes based on Meiri Shabbos 57b to prove like Rav Ovadia that up until 4 centimeters is permitted; See also Ben Ish Chaiy Bo 12 “Women who have a ponytail from behind, one may read Shema opposite them”
The reason: As only a Tefach is considered an Erva by a woman [Igros Moshe ibid]; and less than a Tefach is only Rabbinical, and hence less than majority of a Tefach is allowed. [Rav Ovadia ibid]; See Sefer Tefila Lemoshe [Rav Moshe Levi] 5:50 for a lengthy discussion on this subject.
Opinion of Maharam Alashkar: Maharam Alshiker 35 based on Aruch rules that the sideburns, and other hair that protrudes from the corners of the head covering, is not required to be covered. Some have tried to use this ruling as a source to permit revealing up to a Tefach or two fingers worth of the head. This is not accurate. [See Teshuras Shaiy; Discussed by Rav Yaakov Yosef in Shiur in length]
 Igros Kodesh 9:325 [printed in Shulchan Menachem 6:81] “You want to wear a Sheitel that will leave two finger-worth’s uncovered! Who are you trying to fool, not the world, but only yourself, and what is the purpose of fooling yourself? The rule is clear in Shulchan Aruch [that all the hair must be covered] and there is no room for question on this matter.” See also Igros Kodesh 16:330 and other letters printed in Shulchan Menachem 6:80-84
 All Poskim brought in previous footnotes; So is also conclusion of Rav Yaakov Yosef and Rav Moshe Levi after bringing all the opinions on the subject
 See Tzemach Tzedek ibid
 Zohar Parshas Nasso p. 239, brought in M”A 75:4; Maharam Alshiker 35; Chasam Sofer ibid; M”B 75:14