Gemara

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

138b (Rav Sheshes brei d'Rav Idi): One may wear a felt hat on Shabbos.

Question: It was taught that it is forbidden!

Answer: It depends on whether or not it fits well.

156b: Astrologers had told Rav Nachman bar Yitzchak's mother that her son will be a thief. She told him to always cover his head, in order that fear of Shamayim will be upon him.

One day, his garment fell from his head. He looked up and saw a date tree [of someone else]. His Yetzer ha'Ra overpowered him. He cut a cluster of dates with his teeth.

Kidushin 29b: Rav Huna saw that Rav Hamnuna did not wear a turban like the custom of married men. He refused to look at Rav Hamnuna until he married.

31a (R. Yehoshua ben Levi): It is forbidden to walk erect. "His 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.

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

This is because men sometimes cover their heads, and sometimes not. Women's heads are always covered, and children's are always uncovered.

Rishonim

Rambam (Hilchos De'os 5:6): Chachamim act with great modesty. They do not uncover their heads

Rambam (Hilchos Tefilah 5:5): One may not stand in prayer when his head is exposed.

Sefer ha'Yirah (of R. Yonah): One bends his posture and bows his head, for the Shechinah is above his head. Therefore it is proper to cover his head and not have an uncovered head.

Terumas ha'Deshen (10): Covering the head for Berachos is due to frivolity towards Shamayim, i.e. saying His name without fear. We do not distinguish between in a house and Reshus ha'Rabim.

Terumas ha'Deshen (2:203): The authorities in Breslav enacted that Yisre'elim who must swear must swear by Hash-m's special name like it is pronounced, with their head uncovered. If the authorities intend for their benefit, for they think that this way the oath is severe to the Yisrael and he will not lie, they do not intend to make us transgress. The Rambam and Rosh say that one transgresses to save his life in such a case. If they intend to desecrate the honor of Hash-m's name, surely one must be Moser Nefesh in public. I do not know an explicit Isur of saying His name, with an uncovered head for the need of an oath he is obligated in. Hash-m's name is not l'Vatalah. It is needed to establish the law not to believe their oaths [unless the head is bare]. We do not find an explicit Isur of an uncovered head.

Tashbatz (Katan, 549): There is no Isur to go with his head uncovered. 'I will receive for not going four Amos with my head exposed' was Midas Chasidus.

Poskim

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

Bach (5): The Tur taught that one walks with his head down, and covers his head. He should have taught first covering his head! Rather, he teaches that one need not be careful when sitting. He taught in this order to teach that when one is about to walk, he should cover his head first, lest he transgress walking four Amos with his head uncovered. The Tur is stringent and forbids an erect posture even less than four Amos, for this is a real Isur. If we permit less, perhaps he will go four Amos. Going with his head uncovered is not a real Isur. Rav Huna did not do so. This implies that others were not careful about this! Also in Shabbos, Rav Huna brei d'Rav Yehoshua considered this Midas Chasidus. Therefore, the Tur was not more stringent about this Midas Chasidus than the Amora'im were. The Tashbatz says that covering the head is Midas Chasidus, and the Beis Yosef (91) brings so from Kolbo citing Maharam. This is why he did not bring it in the Shulchan Aruch.

Bach (DH Mihu): R. Yonah connotes that an uncovered head is like an erect posture. It is a total Isur, even less than four Amos. At least it is proper to be careful about an uncovered head due to Midas Chasidus, even less than four Amos. Therefore, he did not mention four Amos. The Rambam connotes like this. In Moreh Nevuchim, he says 'great Chachamim would not uncover their heads, because the Shechinah hovers over man.' He did not distinguish between more or less than four Amos. They (the Rambam and R. Yonah) hold that the Gemara did not say four Amos to permit less than four Amos. There is no difference, for "His honor fills the entire land", and the Shechinah hovers over man.! Rather, this teaches that it is forbidden not only in the market under the sky. Rather, it is forbidden even walking four Amos in his house. Seemingly, we should not be concerned for walking a small amount. Indeed, even less than four Amos is forbidden. This is primary. Even above four Amos is only Midas Chasidus. One must be careful about Midas Chasidus even less than four Amos.

Taz (5): Riding on an animal or in a wagon is like walking, like it says in Kidushin [33b, that if a Chacham rides by, one must rise for him].

Magen Avraham (6): Nedarim 30b connotes that minors normally go with their heads uncovered. In any case it is proper to cover the head so that fear of Shamayim will be upon him, like it says at the end of Shabbos.

Note: Nedarim 30b explained the Mishnah. Had the situation changed by the time the Gemara was written, the Gemara should have mentioned this. Perhaps the Gemara means that children never totally cover their heads, so one can see some of the black [hair].

Darchei Moshe (2, citing Or Zaru'a, brought in Gra DH va'Ychaseh): The Medrash Rabah (Metzora 19:4) expounds "ba'Atzaltayim Yimach ha'Mekareh" - through one's laziness to properly cover his head, the ceiling falters and he is damaged [through scabs].

Mishnah Berurah (10): One may not go [four Amos with his head uncovered] even in a house with a ceiling, and all the more so under the sky all agree that one must be careful.

Mishnah Berurah (11): It is Midas Chasidus to be careful about even less than four Amos, and even when sleeping at night. Some say that even four Amos is not forbidden letter of the law, only for modest people. However, the Taz forbids nowadays from letter of the law, even when sitting in his house. The Magen Avraham says that even minors should cover their heads. The hand suffices to avoid being uncovered. The Pri Megadim says that this is not enough when one is under the sky.

Mishnah Berurah (12): All the more so one may not bless or learn with an uncovered head. His hand does not help, for the body cannot cover itself. Some are lenient in pressed circumstances. E.g. if he wants to drink at night and he has no hat, it suffices to cover his head with his hand. It is better to do like the custom, and put his sleeve on his head. All agree that this is a proper covering. A toupee, even if the hair is sewn on a cloth underneath, is forbidden due to Mar'is ha'Ayin. People will say that it is his own hair. Some are lenient.

Kaf ha'Chayim (15): Some Poskim say that covering the head is merely Midas Chasidus, and some say that it is letter of the law. The Zohar forbids only for Chachamim. For others it is Midas Chasidus.

Kaf ha'Chayim (16): It is Midas Chasidus to be careful about even less than four Amos, and even when sitting. It is permitted for a need, e.g. to cool off.

Kaf ha'Chayim (17): In a bathhouse, there is not even Midas Chasidus. Some are stringent even there (Simlah Chadashah). Birkei Yosef says not to be stringent there. Ben Ish Chai says to cover the head in the outer room, but not in the middle room. One should put the hand on the head even in the inner room when walks four Amos. It does not hurt to do so. Some do so.

Kaf ha'Chayim (18): The Shlah says that it is Midas Chasidus to cover the head at night like during the day.

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

Beis Yosef (DH u'Mechaseh): The custom of pious people is to cover their heads with a turban or Talis, like we find in Kidushin (29b). Surely, Bochurim did not go with exposed heads! Rather, only married men covered with a turban or Talis, for this brings to Yir'as Shamayim, like we find in Shabbos (156b).

Darchei Moshe (4): The Tur says to cover the head with the Talis lest it be uncovered [when blessing on the Talis]. Normally, there is no Isur to go with his head uncovered. It is only Midas Chasidus.

Maharshal (Teshuvah 72): Letter of the law there is no Isur to have an uncovered head, unlike the Terumas ha'Deshen (10) proved [from a Medrash about] Keri'as Shema. Hash-m says 'I did not obligate you to say Shema with uncovered heads." This implies that it is only Midas Chasidus, like it says in Shabbos, and only for going four Amos. However, even when not praying, one should be careful, for people consider this lightheaded and indecent. If the garment [on his head] is heavy while learning, he can cover his head with a light garment or silk. Even if he wants to bless on food at night and he has no hat, or in the bathhouse, it suffices to cover his head with his hand. Regarding separating the heart from the Ervah, the hand does not help, for it is normal to cover his heart with his hand. It does not show that he needs to cover it. A hand on the head shows [that it is for a need]. This suffices, especially for a rare happenstance.

Taz (3): The Tur says to be careful to cover the head [with something else] even when wearing a Talis, for the Talis often slips from the head, and then his head will be uncovered. Nowadays one may not uncover the head at all, since it is a Nochri ritual to uncover the head once they sit - "uv'Chukosiehem Lo Selechu." Further, covering the head shows Yir'as Shamayim (156b).

Gra (6): If the head was covered by a smaller covering, the Tur would not call this 'uncovered.'

Shulchan Aruch (ibid.): Putting one's hand on his head is not a covering.

Support (Taz 3): Putting the hand on the head suffices to avoid the Nochri ritual. It shows that one may not sit with his head uncovered. However, this does not suffice for saying His name, like I wrote below (74:2).

Shulchan Aruch (ibid.): It suffices for Levi to put his hand on David's head.

Igros Moshe (OC 1:1): Ha'Elef Lecha Shlomo requires covering the majority of the head to walk four Amos. The hand suffices even for Berachos, even though it covers only a minority!

Rebuttal (Doleh u'Mashkeh p.2): Also the Chazon Ish requires the majority. We must say that the discussion is about covering the head with both hands.

Shulchan Aruch (301:41): Some forbid wearing on Shabbos a hat to protect from the sun, lest the wind blow it off and he will carry it four Amos in Reshus ha'Rabim, unless the wind cannot blow it off. I.e. it is tight, or his head enters into it, or it is tied with a strap under his chin.

Beis Yosef (DH Kova): One may wear a hat to cover the head, for even if the wind blows it off, he will not carry it in Reshus ha'Rabim, for people do not go with the head uncovered.

Mishnah Berurah (153): One must be careful about a regular hat worn over a Kipah. If the wind could blow it off, he might carry it in Reshus ha'Rabim, since his head will not be uncovered.

Kaf ha'Chayim (241): Eliyahu Rabah permits even with a cap underneath if the practice is not to walk outside without a proper hat.