Gemara

Nedarim 30b (Mishnah): If one vowed from black-headed people, he may benefit from children. Only men are called black-headed.

This is because men cover their heads only sometimes. Children's heads are always uncovered. (One calls them 'people with exposed heads.')

Shabbos 118b (Rav Huna brei d'Rav Yehoshua): I will receive (reward) for not going four Amos with my head exposed.

156b: Astrologers had told Rav Nachman bar Yitzchak's mother that her son will be a thief. She told him to always cover his in order that fear of Shomayim will be upon him, and to request mercy (to be saved from the Yetzer ha'Ra).

One day he was learning under a date tree (not his own). His garment fell from his head. He looked up and saw the date tree. His Yetzer ha'Ra overpowered him, and he cut a cluster of dates with his teeth.

Kidushin 8a: Rav Kahana took a turban in place of the five Sela'im for redemption. To him, it was worth five Sela'im.

(Rav Ashi): Only a great man like Rav Kahana, who needs a turban, may do so.

31a (R. Yehoshua ben Levi): One may not walk erect - "(Hash-m's) honor fills the entire land."

Rav Huna brei d'Rav Yehoshua would not walk four Amos with his head uncovered, because the Divine Presence is above.

33a: A man passed by Ravina and R. Yirmeyah mi'Difti with his head uncovered.

Ravina: That man is so brazen!

R. Yirmeyah: Perhaps he is from the city of Machsiya. People there are used to Chachamim and feel informal with them.

Rishonim

Rambam (Hilchos Tefilah 5:5): One may not stand in prayer when his head is exposed. Chachamim and their Talmidim pray only when they are covered.

Rosh (Berachos 7:35): One may not bless on a Kos Shel Berachah with his head uncovered.

Rosh (Kidushin 1:46): Rav Huna brei d'Rav Yehoshua would not walk four Amos with his head uncovered, because the Divine Presence is above.

R. Tam (in Tosfos Yoma 25a DH v'Ha): The lottery to select Kohanim for Avodah was not done in the Azarah because they would remove the Mitznefes (hat) of a Kohen, and it is improper to stand in the Azarah without a Mitznefes.

Poskim

Shulchan Aruch (OC 2:6): One may not walk four Amos with his head uncovered (Rema - due to honor of the Shechinah).

Shulchan Aruch (8:2): It is proper to cover one's head with the Talis.

Beis Yosef (DH u'Mechaseh): The Tur and Ba'al ha'Itur say 'in order that his head will not be exposed.' They refer to a better cover. One goes to the bathroom before putting on a Talis. Surely, his head is not exposed!

Rebuttal (Gra 8:6): The Beis Yosef forbids walking with an uncovered head. This is wrong. It is Midas Chasidus not to. It is improper only in front of a Chacham. Kidushin 8a connotes that regular people do not require a turban. The Tur and Ba'al ha'Itur mention covering the head with a Talis in order to pray, for then everyone should cover his head. In summary, the only time it is forbidden for one's head to be uncovered is in front of Chachamim. It is very proper to cover it at the time of prayer. Kedoshim should always cover it.

Darchei Moshe (4): One should follow the Beis Yosef's words. However, it seems that the Tur requires covering the head when putting on a Talis. At other times it is merely Midas Chasidus.

Taz (8:3): Nowadays it is a Nochri ritual to bare the head immediately upon sitting down. Therefore, "uv'Chukoseihem Lo Selechu" forbids this, especially because covering the head shows Yir'as Shamayim.

Question (Darchei Moshe 2): The Kol Bo calls this Midas Chasidus. R. Peretz forbids going to a Beis ha'Keneses with an exposed head. The Beis Yosef forbids. The Tur disagrees. Siman 91 forbids only at the time of prayer!

Eshel Avraham (6): Perhaps the Beis Yosef holds that one wears a small covering to the Beis ha'Keneses, and removes it before entering. Or, he covers his head with his hand, which does not suffice in the Beis ha'Keneses.

Magen Avraham (6): It is Midas Chasidus not to go even less than four Amos (Maharshal Teshuvah 72). Nedarim 30b connotes that children go with uncovered heads. Nevertheless, it is proper to cover them in order that they will have Yir'as Shamayim, like we find regarding Rav Nachman.

Eshel Avraham (6): It seems that even in a roofed house it is permitted only to sit still, but not to walk.

Taz (5): One may not ride this distance, for riding is like walking.

Shulchan Aruch (91:3): Some forbid saying Hash-m's name when his head is exposed. Some forbid entering a Beis ha'Keneses when his head is exposed.

Beis Yosef (DH u'Mah she'Chosav): The Kol Bo derives from Shabbos 118b that it is mere Midas Chasidus not to walk with one's head uncovered. R. Peretz forbids entering a Beis ha'Keneses when his head is exposed. There are two opinions in Maseches Sofrim (14:15) about whether or not one may bless when his head is exposed. R. Yerucham forbids, so we are stringent.

Rebuttal (Maharshal Teshuvah 72): I know of no source to forbid blessing with an uncovered head. R. Yerucham forbids like the stringent opinion in Maseches Sofrim, but Vayikra Rabah (27:6) says 'Hashem did not burden us to say Kri'as Shma with uncovered heads.' This implies that it is permitted!

Gra (8:6): The stringent opinion in Maseches Sofrim is only Midas Chasidus. Berachos 60b and Shulchan Aruch (46:1) support this. (They list several Berachos one says in the morning before putting on a turban. We would do so nowadays if not for Tum'as Yadayim or Amei ha'Aretz.)

Question (Mishbetzos Zahav 8:3): One may not say Shma with an uncovered head, because it contains Hash-m's name!

Kaf ha'Chayim (12): The primary opinion is stringent. One must be stringent even about saying His name in other languages.

Maharshal (ibid.): I cannot fathom why people are stringent even not at the time of prayer. It is forbidden only for women (Kesuvos 72a), or Midas Chasidus not to go four Amos with an uncovered head. Kidushin 31a forbids walking erect, and says that Rav Huna would not walk four Amos with his head uncovered. He did not forbid this! Nowadays, people walk erect, but they are careful to cover their heads as if the Halachah required this! We are concerned for an uncovered head only under the sky. However, Midas Chasidus not to say Hash-m's name with an uncovered head applies even in the house. The Ram (mi'Rotenberg, Tashbatz 549) calls all this Midas Chasidus. The Tur requires covering the head during Tefilah, but not during Kri'as Shma. However, people consider this to be forbidden, so I cannot permit it. A Chacham should be careful not to learn with his head uncovered, because people consider this to be forbidden.

Shulchan Aruch (5): One may not stand in prayer when his head is exposed.

Shulchan Aruch (6): It is the practice of Chachamim and their Talmidim to pray only when they are covered.

Shulchan Aruch (151:6): Some forbid entering a Beis ha'Keneses with an uncovered head.

Bi'ur Halachah (DH Oh): This is forbidden even in a place where people go in front of nobles with uncovered heads.

Rema (282:3): One may not read from the Torah with his head exposed.

Magen Avraham (8): This refers to a minor. An adult's head may not be uncovered in any case (2:6 and 151:6). Yoma 25a supports this.

Kaf ha'Chayim (27): If he reads Hash-m's name, or is in a Beis ha'Keneses, in any case his head must be covered! Here we forbid even if he does not read Hash-m's name, and even in a room outside the Beis ha'Keneses. Alternatively, we require a big hat due to honor of the Tzibur. Normally, a small cap suffices.