Is it common for a mathematics Master's student to have no publications?

I cannot speak for the whole world, but for a master student in Europe, it would be exceptional to have a published paper already. A preprint is perhaps not completely unheard of but still extremely uncommon.

Don't worry about it. Having a publication on entry to a (US) Ph.D. is unusual (though not unheard of), regardless of whether you're coming in with a master's or a bachelor's degree. In fact, in most subfields of mathematics, most of those completing a Ph.D. will not yet have a paper submitted, much less accepted or published (though many will have turned all or part of their thesis into a preprint by then.)

That being said, for admission to a competitive Ph.D. program, it is very helpful to show some evidence of excellence in mathematics beyond coursework. A publication/preprint is one way to do that; good competition results in math contests another; great reference from a professor a third, etc. etc. So figure out what is your excellence calling card.

Good luck!

In the US, there are diffent kinds of masters degrees. Some are mostly course based and some have a research component. Of those, not all will require formal publication outside the university. This guides the decisions of those on admissions committees for doctoral programs.

Having a publication in a good journal is a strong plus, but lacking one isn't necessarily a strong minus. At the top schools it would be likely to count for more than otherwise, but still, not likely to be essential.

I suspect that for students applying it is not especially common to already have publications. This is partly because of the short and time-limited nature of most US masters programs. Research tends to be open ended, hence the long time it can take to get a doctorate.