Taking off a Kippa - Igrot Moshe Yoreh Deah 3:2

In a teshuva from 1974, Rav Moshe was asked about whether it would be permitted for a person to remove his kippa in a business situation if keeping it on would cost him financially.

Rav Moshe replied that according to virtually all poskim wearing a kippa is a מידת חסידות, and even though it has been widely accepted by Jews it nevertheless takes a back seat in the face of potential financial loss, which qualifies as אונס. The one possible difficulty is the view of the Taz, who raises the possibility that not wearing a kippa would be considered חוקות הגוים. However, Rav Moshe rejects this view based on the fact that nowadays, a man walking around with a bare head is considered de rigeur from a fashion standpoint and is not done for any ritual purposes. As such, a Jew walking around with a bare head would not be considered to be conforming to any specifically non-Jewish practice, and when money is on the line he should remove his kippa rather than risk his income.

Small Kippot - Igrot Moshe Orach Chayim 1:1

In this first teshuva in Igrot Moshe, Rav Moshe Feinstein was asked if one is permitted to walk around with his head only partially covered, i.e. if he is wearing a small kippa.

Interestingly, Rav Feinstein does not attempt to define how much of the head is too much or too little (at what point in the slope downward of the head does one no longer need to cover it?). However, he cites the view of Rav Shlomo Kluger who states that one can walk fewer than four amot with the head partially covered and not at all if the head is uncovered. Rav Feinstein's analysis then demonstrates that Rav Kluger seems to have an extreme view on this point, and that the wearing of a kippa is largely a מדת חסידות, although nowadays it has been accepted by Jews as a requirement for everyone. Rav Feinstein concludes that given its near-universal acceptance, one should not even sit without wearing a kippa, as there may be a חוקות עכו"ם problem (although it is not completely clear what practice of the non-Jews this would be emulating).