As the title suggests, I am a Master's student with no publications. I will apply for a PhD in the USA next year so I am a bit worried that most applicants with a Master's degree have a publication. Is this (no publications in Master's) common within mathematics?
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.
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.
No, a Master's degree is a "post baccalaureate," not a "pre PhD.
Many master's programs have a "capstone" requirement consisting of a paper or a project, but it is uncommon for it to be of publishable quality.
Basically, you are usually not qualified to do "research" until toward the end of the of PhD program, and your thesis may well be your first publication.
It is very unusual to have a paper -- further, it would be extremely unusual to have a solo-author paper, and whether you have a paper with one of the project supervisors you had in your undergraduate or Masters depends more on them that it does on you.
Having a paper is a (small) positive, but not at all a requirement. Appearing interested and knowledgable at your interview is much more important.