Gemara

(Mishnah - R. Eliezer): One is liable for tearing out [by hand] hair from his head, mustache, or beard;

Chachamim forbid only mid'Rabanan.

81b (Mishnah): A Nazir may be Chofef (rub or shampoo) or Mefasfes (ruffle) his hair, but he may not comb it.

Nazir 42a - Question: The Reisha permits rubbing or ruffling. This is like R. Shimon, who permits Davar she'Eino Miskaven (something unintended that may result in a transgression, i.e. he might uproot hairs);

The Seifa forbids combing. This is like Chachamim!

Answer (Rabah): The entire Mishnah is like R. Shimon. One who combs intends to uproot dangling strands.

(Mishnah - R. Yishmael): He may not be Chofef with earth, because it causes hair to fall out.

Rishonim

Rambam (Hilchos Nezirus 5:14): A Nazir may not comb with a comb, for this definitely uproots hairs.

Rosh (Beitzah 2:24): The Gemara discusses whether combing animals wounds them. We see that scraping horses with iron combs uproots hairs. This is a Pesik Reishei, like it says regarding a Nazir! We must say that their combs were unlike ours. (It was not a Pesik Reishei.)

Beis Yosef (OC 523 DH Mekadrin): Tosfos (Beitzah 23a DH v'Shalosh) says like the Rosh. R. Yerucham says that some permit even with our combs. Since he is not pleased with detaching the hair, it is not a Pesik Reishei, like we say about like Nizyasa (Kesuvos 6a), that one may cram a piece of cloth into the opening of a barrel of beer to seal it [on Yom Tov, according to R. Shimon]. This is unlike combing hair, from which one benefits.

Rashi (81b DH Aval Lo): A Nazir may not comb, for this definitely uproots hairs. R. Shimon agrees that this is forbidden.

Tosfos (42a DH Aval): The Gemara connotes that Stam, one who combs intends to remove [dangling] hairs.

Rashi (42a DH Lehashir): Since he intends to remove dangling hairs, this is Davar she'Miskaven, and it is forbidden.

Rivash (394 DH Ra'isicha, b'Emtza Devarav DH b'Inyan): The Gemara in Nazir connotes that combing is not a Pesik Reishei to uproot hairs, but it is forbidden because he intends to uproot dangling hairs. This applies to a Nazir, who will shave off all his hair. All the more so a Stam person is concerned for dangling hairs! The Gemara said that anyone who combs intends for dangling hairs, not only a Nazir! We are more stringent about Shabbos than regarding Nazir. We decree about looking in a metal mirror on Shabbos (lest he use it to cut dangling hairs - Shabbos 149a), but not regarding a Nazir. Also, a Nazir transgresses a Lav only if he cut the hairs [nearly] as close a razor, so that the end cannot be bent to the root. The Melachah of shearing on Shabbos applies even if he left much attached, and there is Skilah and Kares. Rashi says that it is a Pesik Reishei. This is difficult. Perhaps it is because combing was taught between Chafifah with niter and sand, and Mefasfes, and Chafifah with earth, which R. Shimon forbids due to uprooting hairs. Rather, it is all due to a Pesik Reishei. The questioner in the Gemara thought that combing is not a Pesik Reishei if one's hair is arranged and there are no dangling hairs. Rava answered that it is a Pesik Reishei, for one combs only if there are dangling hairs, in order to uproot them. Combing is only for uprooting dangling hairs and hairs tied to each other, and they are cut or uprooted through combing. People do not comb needlessly! Rashi in Shabbos (94b) says that fixing hair with a comb is Asur due to Binyan (building). He did not say 'due to shearing, for it is a Pesik Reishei'! There, it does not mean combing, rather, braiding. Or, it is passing a comb wet with oil and water on a combed head to make the hairs stick together.

Kaf ha'Chayim (OC 303:122): Also Chidushei ha'Ran (94b) says so in the name of the Ro'oh. However, Tosfos says that this depends on the argument [of R. Eliezer and Chachamim].

Poskim

Shulchan Aruch (OC 303:27): One may not comb with a comb on Shabbos, even if the comb is made of pig's hairs, for one cannot avoid uprooting hairs.

Beis Yosef (DH u'Mashma): We hold like R. Shimon, who permits Davar she'Eino Miskaven. All forbid combing with a comb, for one who combs intends to uproot dangling strands, like it says in Nazir. This Mishnah is brought in Shabbos 81b regarding Shabbos. Kolbo says that we do not pass a comb over the head, even [a comb of] an elephant, and even a comb of pig's hair, for one cannot avoid uprooting hairs.

Beis Yosef (DH v'Shamati): Some say that according to those who hold that R. Shimon permits a Pesik Reishei d'Lo Nicha Lei (one is not pleased with the result), one may comb, for he does not want to uproot hairs. I reject this. He wants his hair to be even and nice. This necessitates removing some hairs, so it is a Pesik Reishei d'Nicha Lei. R. Yerucham connotes like this regarding scraping a horse on Yom Tov. Some permit, even though it uproots hairs. Since he does not benefit, it is not [forbidden like] a Pesik Reishei. This is like Nizyasa. It is unlike combing hair, for there he benefits. Nidah 67b forbids a woman to do Chafifah on Shabbos. Chafifah is combing, like Rashi says in Bava Kama (82b). Semag and Semak say so. This is why if one will immerse on Motza'ei Shabbos, she combs on Erev Shabbos. The Rivash (394) brought additional proofs that one may not comb on Shabbos.

Magen Avraham (22): Rashi in Shabbos says that it is a Pesik Reishei (that one will uproot hairs). Perhaps Rashi comes to teach that even if one does not intend to uproot hairs, it is forbidden. The Gemara discusses a soft comb. The Rivash says that girls use a Kli from pigs' hair. It is known that it does not uproot hairs. We permit on animals, which have thick hair Combing human hair is forbidden, even according to the opinion that permits a Pesik Reishei d'Lo Nicha Lei, because he wants to separate hairs, which necessitates uprooting hairs. One is liable for shearing, just like for cutting nails. In the Mishkan they sheared skins even though they did not need the wool.

Magen Avraham (23): Rashi (50b DH Chofef) says that Mefasfes is Menafetz (spreading out). Tosfos and Rashi in Nazir say that Mefasfes is separating hairs from each other. The Rivash connotes that passing a wet comb on a combed head is forbidden only due to Binyan.

Shulchan Aruch (ibid.): One may do Chafifah and be Mefasfes by hand.

Gra (DH Asur): The She'altos says that Chafifah is forbidden on Shabbos because one may not comb on Shabbos.

Mishnah Berurah (86): It is forbidden because it is a Pesik Reishei. The Gemara says that anyone who combs intends to uproot dangling hairs. This forbids even with a soft comb, in which case it is not a Pesik Reishei, unless he does not intend to uproot them. Yeshu'os Yakov bewails that people comb, which is Chilul Shabbos; Chachamim should protest, for perhaps people will listen. Do not say that one must comb, lest he be disgraced. There is a permitted way (with a soft brush, without intent for dangling hairs). Also, one may do Chafifah and be Mefasfes by hand!

Mishnah Berurah (87): The custom is to designate the brush from pigs' hair for Shabbos, so it will not be Uvda d'Chol (a weekday action).

Kaf ha'Chayim (123,124): We forbid a comb from pigs' hair, even though the teeth are soft, but not a brush from it.

Kaf ha'Chayim (126): Even what is almost certain to occur, is forbidden like a Pesik Reishei.

Bi'ur Halachah (DH Lachuf): The Gemara forbids Pokeses due to Shevus. It is like Binyan, just like braiding. Rashi says that she fixes her hair with a comb or her hands. The Rivash says that he does not refer to combing, for it is forbidden due to shearing. Also, the Gemara did not say that she combs with a comb. Rather, it is another way of fixing hair with a comb. Perhaps it is passing a wet comb over a combed head. The Magen Avraham holds that the Rivash forbids using a comb to fix hair in any way. However, perhaps this is only if the head was not already combed, therefore any fixing will uproot hairs.

Question (R. Akiva Eiger, Shevuos 2b DH v'Ayen): Tosfos (DH Chayav) forbids Hakafah (removing the sideburns) using tweezers. Why may one comb the Pe'os? It is a Pesik Reshei that this will uproot hairs! This requires investigation.

Answer (Chasam Sofer 2:139): A Mishnah forbids a Nazir to comb his hair. This implies that other men are permitted! This shows that we are not concerned for two stringencies together, i.e. to forbid cutting sideburns without a razor, and to forbid even two hairs.