What Can One Do without a Kippah?
- One should not walk (or travel) 4 Amot without a Kippah. Nowadays, the Kippah serves as an identification of religious Jews and so it is important to wear it in order to show that one fears Heaven. 
- It is a proper and praiseworthy practice to wear a Kippah even when one walks less than 4 Amot  or one stands or sits for the time it takes to walk 4 Amot , whether one is outside or indoors. 
- One is not allowed to make brachot without a Kippah, but if by accident (for example, the kippah fell off and one did not notice) one made a bracha without a Kippah, the bracha is acceptable after the fact.  Others believe one is require to repeat the bracha. 
- One is allowed to think about Torah even if one is not wearing a Kippah, such as if one is in bed (although not if one just awoke and has yet to say Birchot HaTorah) or by a pool. 
- One is allowed to greet a Jew who is not wearing a Kippah even if he will respond “Shalom” which has the status of Hashem’s name. 
- One is permitted to shower, sleep, and swim without a head covering. Even at places where the minhag is not to wear a Kippah, such as one who is swimming, one must nonetheless wear a Kippah in order to say a bracha. 
- Riding in a moving car, train, or plane is considered like walking. 
Wearing a Kippah to Sleep
- There is a pious practice to wear a Kippah to sleep. One need not be concerned about it possibly falling off whilst asleep. 
Size and Material of Kippah
- One should wear a Kippah that is recognizable from all sides (front, back, and sides), but from Midat Chasidut one should wear one that covers the majority or the entire head, especially when one is saying Kriyat Shema, Shmoneh Esrei, and Birkat HaMazon. 
- A Kippah with holes in it is an acceptable Kippah. 
- If someone is wearing a wig with which there is a cloth underneath that’s not visible to the outside, there is a dispute whether it counts as a Kippah or not. It is preferable to be strict especially during Tefilah and while making brachot. 
- Putting one’s hand on one's head does not suffice to be able to make a bracha, one's hand is not considered a Kippah. however someone else’s hand or one’s sleeve is sufficient. 
Who Is Obligated in Wearing a Kippah?
- Children should wear a Kippah to inspire Yirat Shamayim.  It is forbidden for even a small boy to recite any prayers or blessings if his head is uncovered. 
- The minhag is that unmarried women do not cover their heads, yet it is correct for them to wear a head covering during Shmoneh Esrei. Those who do not wear a covering at all have what to rely on. 
Wearing a Kippah in a Place not Suitable for a Religious Jew
- If one is in a place that is not suitable for a religious Jew to be in such as a party with mixed dancing (or perhaps even a museum or theater) one should not remove one’s Kippah so as not to make a Chilul Hashem. 
Wearing a Kippah at Work
- If one cannot get a job unless one does not wear a Kippah at work, one does not have to forfeit one's job for this mitzvah. If they allow one to wear a regular hat one must wear such a hat. Also, when one enters another room or the marketplace one must put ones Kippah back on even if one will be mocked, as long as there is no concern of losing one’s job. 
- Gemara in shabbat 156b
- Taz 2:5 writes that traveling is the same as walking in this regard. Malbim in Artzot Hachaim (6, Meir LeAretz 48) and Halacha Brurah 2:11 concur.
- The Gemara Kedushin 31a records Rabbi Yehoshua’s practice not to walk 4 Amot without a head covering so that he would be reminded of the Shechina that is above him. Shabbat 156b writes that Rav Nachman’s mother made sure her son always wore a head covering so that he would always be cognizant of the fear of Heaven.
- There is a long standing dispute whether there is an obligation or whether it is only a Midat Chasidut to wear a Kippah. The Zohar (Pinchas pg 245b, Naso pg 122b) implies that it is an obligation for a Talmid Chacham but for everyone else it is only a Midat Chasidut to wear one. Rambam (Deot 5:6; Moreh Nevuchim 3:52), Kol Bo 11, Orchot Chaim (Tefiliah 48), and Tashbetz 547 quoting the Maharam imply that it is only a Midat Chasidut for a non-Talmid Chacham. However, Sefer Manhig (Tefilah 49), Kitzur Piskei HaRosh (first perek of Kedushin), and Sh”t Mahari MeBruna 34, 165 write that it is a obligation on every Jew to wear.
- S”A 2:6 writes "It is forbidden to walk with an arrogant posture and one should not walk 4 Amot without a head covering". The language of Shulchan Aruch implies that wearing a Kippah is only a Midat Chasidut. See also Bet Yosef (Siman 8, 46, and 91). Many poskim agree that it is only a Midat Chasidut including the Darkei Moshe (2:3 and 8:4), Maharshal 72, Birkei Yosef 2:2, Magen Avraham 91:3, Buir HaGra 8:6, and Maamer Mordechai (2 and 91:5).
- However, the Taz 8:3 writes that it is forbidden not to wear a Kippah because of Chukat Akum. (Even according to this opinion one can be lenient if one has a reason to take off the Kippah as per Bet Yosef Y”D 178 and Rama against the Gra who forbids Chukot HaGoyim even if there is a reason to take it off.) Pri Megadim A”A 2:6 and Sh”t Elef Lecha Shlomo O”C 3 differentiate between a complete head covering which is a Midat Chasidut and a partial head covering which is an obligation. However, Sh”t Igrot Moshe O”C 1:1 argues on this distinction and adds that perhaps one can make such a distinction within the opinion of the Taz. Rabbi Shalom Mashash in Shemesh U'Magen 2:58 writes that when walking outdoors or in a shul it is obligatory. Sh”t Otzrot Yosef 1:4, Sh”t Yabia Omer 6:15 (4-5), Sh”t Yachave Daat 4:1, and Yalkut Yosef 1:7 write that even if the halacha is that it is only a Midat Chasidut, nowadays, since wearing the Kippah is a symbol of being a religious Jew, wearing a Kippah is somewhat more obligatory than a Midat Chasidut because there is a concern of Marit Ayin (suspicion) if one is seen without a Kippah.
- Sh”t Mahari MeBruna 34 permits it if one walks less than 4 Amot. However the Bach 2 infers from Rambam (Deot 5:6; More Nevuchim 3:52) that less than 4 Amot are also forbidden. Taz 8:3, Bchor Shor (Shabbat 118b), and Birkei Yosef 2:3 concur. Magen Avraham 2:6 writes that it is only a Midat Chasidut to wear it for walking less than 4 Amot.
- Sh”T Mahari MeBruna 34 permits if one is just sitting and Magen Avraham 282:8 permits whether one is sitting or standing. Bechor Shor (Shabbat 118b), Birkei Yosef 2:3, Halacha Brurah 2:11 are strict as long as one waits the time it takes to walk 4 Amot.
- Sh”t Maharshal 72 permits one not to wear a Kippah indoors. Knesset Hagedolah 2, Bear Heteiv 2:5, and Olat Tamid 2:5 quote the Maharshal. However, the Bach 2 argues on this distinction. Eliyah Rabba 2:4, Mishna Brurah 2:10, and Halacha Brurah 2:11 concur.
- Sh”t Otzrot Yosef 1:5 say that since the Rosh (on Brachot 60b), Rambam (Tefilah 7:4), and S”A 4 hold that the order of the brachot is precise and Oter Israel BeTifarah is made for having a head covering, implying that the other brachot can be made without a Kippah (Gra 8:6 makes similar implication from Rif). Nonetheless, Masechet Soferim has a dispute whether one can say Hashem’s name without a Kippah and Rabbenu Yerucham (quoted by Bet Yosef 91:3), Or Zaruha 2:43, S”A 91:3 rule stringently. Lechem Yehuda (Tefilah 5:5) argues that it seems that S”A 91:5 holds that one one is required to wear a Kippah when saying Shmoneh Esre. Perhaps S”A (retracting from his ruling in Bet Yosef like Rabbenu Yerucham) rules with the term “Yesh Omrim” and then an anonymous opinion and so we should follow the anonymous opinion. Yet, Sh”t Yabia Omer O”C 6:15(3) rejects this because S”A 206:3 is very clear that one may not say Hashem’s name without a Kippah and S”A 91:5 meant there is an added reason to wear a Kippah in Shmoneh Esre. Sh”t Maharshal 72 says that perhaps it is permitted according to halacha, but since the Minhag is not to say a bracha without a Kippah one should not be lenient (Chida in Sh”t Chaim Shaal 2:35 and Kiseh Rachamim (Masechet Soferim 14) write similarly). Gra (Biur HaGra 8:6 and Meorei Or (Bear Sheva 15b)) writes it is only a Midat Chasidut and if there is a bracha which one will miss if one gets a Kippah (such as if he just heard thunder) one can make the bracha without wearing a Kippah. Sh”t Yabia Omer 6:15(6) says bedieved one has fulfilled his bracha. Rav Shlomo Zalman Auerbach (Halichot Shlomo Tefilla 2:16) agrees that one need not repeat it.
- Sh"t Iggerot Moshe OC 4:40:14
- Sh”t Yabia Omer 6:15(7), Sherit Yosef 2 pg 370
- Sh”t Yabia Omer 6:15(8). See further Sh”t Otzrot Yosef 1:5e
- Sh"t Yabia Omer 6:15:7, Sh"t Otzrot Yosef 1:5
- Vezot Habracha (p. 8)
- The Beeir Heiteiv 2:6 quotes from the Taz that one must wear a Kippah when riding in a cart or on an animal.
- Sh”t Otzrot Yosef 1:5 say that since the Rosh (on Brachot 60b), Rambam (Tefilah 7:4), and S”A 4 hold that the order of the brachot is precise and Oter Israel BeTifarah is made for having a head covering, implying that the other brachot can be made without a Kippah (Gra 8:6 sees this similarly implied in the Rif). Certainly then, there is no obligation to wear a Kippah to sleep as the Leket Yosher pg 46 writes. However, Eliyah Rabba in the name of the Shlah says that one should wear it to sleep as a Midat Chasidut. Mishna Brurah 2:11, Kaf HaChaim 2:18, and Halacha Brurah 2:12 bring this opinion as halacha.
- Halacha Brurah 2:12 (quoting Sh”t Otzrot Yosef 1:4e), Sh"t Yechave Daat 4:1. Pri Megadim A”A 2:6 and Sh”t Elef Lecha Shlomo O”C 3 differentiate between a complete head covering which is a Midat Chasidut and a partial head covering which is an obligation. However, Sh”t Igrot Moshe O”C 1 argues on this distinction and adds that perhaps one can make such a distinction within the opinion of the Taz. Sh”t Otzrot Yosef 1:5 mentions that there is an added concern by Kriyat Shema, Shemona Esrah, and Birkat HaMazon as in Mishna Brurah 91:9 (concerning Shema) and brachot 51a (concerning Birkat HaMazon).
- S”A 91:4 based on Sh”t Trumat HaDeshen 10 says that even a straw head covering is acceptable. Sh”t Chatam Sofer 6:2 (quoted by Halacha Brurah 2:14) says that this permits even Kippot made with holes in them.
- Pachad Yitzchak (20 s.v. Kama Chasif, 80 s.v. Peirukanu), Olot HaTamid 2, Sh”t Levushei Mordechai (Tanina O”C 108) say that Marit Ayin applies to Kippah and so one should wear a Kippah on top of the wig. Maamer Mordechai 91:6 limits it to Tefilah and brachot. However Mekor Chaim 2:6, Hagahot Rabbi Akiva Eiger 91, and Artzot HaChaim (2, Meir LeAretz 54) argue that there is no issue of Marit Ayin since Kippah is only a Midat Chasidut to start with. Mishna Brurah 2:12 quotes the dispute without ruling on the topic. Sh”t Chaim Shaal 2:35(1) writes that one should wear it because of Minhag. Halacha Brurah 2:15 quotes the dispute and says it is preferable to be strict especially during Tefilah and brachot.
- S”A 91:4 (based on Sh”t Trumat HaDeshen 10 against the Sh”t Maharshal 72 who is more lenient) rules that one’s own hand is not acceptable but one’s friend’s hand is acceptable. Eliyah Rabba 91:5, Taz 8:3, Machsit HaShekel 91:4, Artzot HaChaim 2:6, Mishna Brurah 2:11, Halacha Brurah 2:16 conclude that one can rely on the Maharshal to walk four Amot but not to make brachot. Bach 91 says one’s sleeve is acceptable and that is the Minhag we follow. Mishna Brurah 2:12 and Halacha Brurah 2:16 bring this as halacha.
- Magen Avraham 2:6 proves from the Gemara that a child does not need a Kippah but it is correct for them to have a Kippah to inspire Yirat Shamayim. Eliyah Rabba 2:4, Shulchan Aruch HaRav 2:7, Kitzur Shulchan Aruch 3:6, Mishna Brurah 2:11, Artzot HaChaim 6, and Halacha Brurah 2:19 concur.
- Children in Halacha pg. 14
- Yalkut Yosef (91:8, Tefilah pg 318, Sherit Yosef 2 pg 368), Sh”t Otzrot Yosef 1:5, Tzitz Eliezer 12:13, Tefilla KeHilchata quoting Echad MeGedolei HaDor
- Sh”t Igrot Moshe O”C 2:95e writes that if one goes to a museum or theater which is forbidden, one should not take off one’s Kippah, we do not say since you are doing one sin you can do another also. [Perhaps Rav Moshe considers entering a museum or theater inappropriate, because of possible inappropriate or profane material that is on display.] A more substantial claim is that if one’s Yetzer Hara is overpowering and one is going to go to the museum anyway, in order to minimize Chilul Hashem one should take off one’s Kippah if one does it Leshem Shamayim. Rav Moshe counters this by writing, that if we are speaking about a person giving into their Yetzer there is little chance he is going to take off his Kippah Leshem Shamayim and so it is forbidden. Sh”t Igrot Moshe Y”D 2:33 writes that teens going to a place where there is mixed dancing and Peritzut which is totally forbidden, should not remove their Kippah. He says that the claim of "if they wear their Kippah people will think that a religious Jew is allowed to be involved in mixed dancing" is incorrect, we do not tell someone who is already doing a sin to do another one. Following the incorrect logic, Rav Moshe writes, we should also tell him not to follow any mitzvah so it is clear he is a Rasha so that people will not think that a religious Jew can dance in a mixed setting. Since he is doing a sin in public it is also a source for Chilul Hashem but would only increase Chilul Hashem for him to do another sin of not wearing a Kippah. Nonetheless, the religious community should make it clear that it is forbidden, so that people do not learn from such teens who are sinning.
- Sh”t Igrot Moshe (C”M 1:93 and O”C 4:2) writes that since wearing a Kippah is neither a positive or negative mitzvah one need not lose a large portion of one's money by not taking such a job. However since they do not care if you wear it in another room or in the marketplace one must wear it in such places. Sh”t Igrot Moshe Y”D 4:11(3) adds that even according to the Taz who holds there is a prohibition of Chukot HaGoyim (that the practice of Goyim used to be to always wear hats and when they sat down to eat or something similar they would take off their hat) nowadays it would not apply since people do not generally wear hats. Thus, one can take a job at a place where they do not allow one to wear a Kippah. However, if they allow one to wear a regular hat (not a Kippah) one should wear a hat.
- It is worth noting that the US Federal Government and the US Army have recently made it acceptable to wear a Kippah while in service (Source).