There is one time when a head covering is required as is brought in מסכת סופרים, and that is when one takes out the Torah scroll and reads it in the synagogue,-- and even that is never brought down in the gemara nor in the later poskim. So it is hard to know why people get so frantic if you walk into a shul without a head covering. It  has no source in halacha. It is just a sign of a deeper problem,  -- that  nothing they do has any source in Torah. The only people that are making an effort to keep Torah as it says [and not add or subtract] are the Litvaks that follow the path of the Gra and Rav Shach.

[In pictures of Jews in the Middle Ages, there is not one showing any Jew wearing a "Kipa". The only people who wore "kipot" were the pope and bishops.-as they do today.