Outside of prayer/worship, where is it written that there is a requirement to wear a kippa?

Question

Outside of prayer/worship, where is it written that there is a requirement to wear a kippa?

I have heard that there's no halachic requirement to wear a kippa other than for prayer/worship.

Answer 1

Firstly in the Shulchan Aruch, Orach Chayim 2:6 it writes:

אסור לילך בקומה זקופה ולא ילך ד' אמות בגילוי הראש ויבדוק נקביו

It is forbidden to walk in an [overly] upright posture, and one should not walk [a distance of] four amot with the head uncovered (out of respect for the Divine Presence), and one should examine one's orifices [to ensure they are kept clean]. (Sefaria Translation)

The Mishna Berurah helps develop this idea further:

ד"א - ומידת חסידות אפילו פחות מד"א ואפילו בעת השינה בלילה ויש שמצדדין לומר דאפילו ד"א אינו אסור מדינא רק להצנועין במעשיהן אבל כבר כתב הט"ז לקמן בסימן ח' דבזמנינו איסור גמור מדינא להיות בגילוי הראש ואפילו יושב בביתו עי"ש הטעם וכן כתב בתשובת מהר"י ברונא וכתב המ"א דאפילו קטנים נכון להרגילם בכיסוי הראש כי היכי דליהוי להו אימתא דשמיא כדאיתא (בשבת קנ"ו) כסי ראשך כי היכי דליהוי עלך אימתא דשמיא ודע עוד דלענין גילוי הראש די בכיסוי היד על הראש...

4 amot - And it is pious trait (to wear a kippa) even less than 4 amot and even at time when one sleeps at night. And there are those that are on the side to say that going even 4 amot is not forbidden from the point of law but rather to be modest in their deeds - the Taz already wrote earlier in Siman 8 that in our time it is completely forbidden to have one's head (lit. revealed) uncovered and even to sit in his house (see over there for a reason ). Similarly it writes in the responsa of the Mahari Bruna, and the Magen Avraham writes that even young children, it is correct for them to regularly cover their heads as how then will they gain a fear of heaven? Like it says in Shabbos 156 "Cover your head so that the fear of Heaven will be upon you." And furthermore, with regard to one's head being uncovered, it suffices to cover with one's hand on their head...

So we see that covering one's head with a Kippa / Yarmulke extends well beyond only prayer services.