One of my favorite pieces of Shulhan Arukh is the following:
לא תלכנה בנות ישראל פרוענות בשוק, אחת פנויה ואחת אשת איש (Even HaEzer 21:2)
The daughters of Israel should not walk with uncovered heads in the marketplace, both unmarried and married women.
It has always sounded pretty definitive, to me. In fact, when I was in college, I had fairly strong feelings that I was "doing it right" in contrast to my Orthodox peers, in this matter. Of course, nothing stays that simple- halakha expands and comments and interprets, and as you'll see later, the term פנויה/unmarried woman gets interpreted in rather surprising ways in this case. I don't want to jump ahead of myself, but the interpretations lead to what is normative practice today, which very clearly does not involve all Jewish women wearing a head-covering of some sort. Although, before I was married, this source was one of the ones that gave me support in choosing to wear a kippah or the equivalent all the time, without feeling like it was inauthentic.
I'm also told, although I haven't researched it myself yet, that there are Sefardi/Edot HaMizrach communities where unmarried women do, or at least did, cover their heads. And I've heard of Sefardi rabbis who promote the idea that all women should have a head-covering of some sort to pray. It'll be an interesting area to explore... I expect that the fact that women in the broader society in which those Jewish communities were living did cover their heads influenced this halakha and its interpretation.