Can I say I'm undecided yet in my statement of purpose ? (PhD in math) I just started my 2nd year of master and it's already time to apply for a PhD in math, deadlines are in ~1+ month.

In most samples, students had a clear idea of what they would like to study. They write that they have studied this and that, and that they would like to be under Prof. X's supervision because they worked on his papers.

As for me, I liked most math topics that I have studied, with very few exceptions. In short, I'm undecided, exactly when I buy an ice cream and  I'm facing 20 good flavours. 

Most universities, on their application page, offer "undecided" among the options (algebra, number theory, ...). Although they may say "it's totally fine" (because everything is nowadays), what is the truth behind the appearances?

Stories from people who are on admissions committees or from [un]successful applicants who wrote that in their SOP are most welcome.

EDIT: I successfully got admitted to some really good US universities with "undecided" in my SOP
Generally in the US you don't need to choose an advisor when you start Doctoral study. Therefore, having a firm idea about a concentration, research focus, is less important than in some other places. But giving the admissions people some idea will give the a better way to think about how you might fit in. 

What I would do, in your case, is to say you are undecided, but currently focused on X and Y, where X and Y are a couple of things you know a lot about. It narrows the focus a bit without limiting you in any way. But, having an MS means that you will want to get started on research pretty soon after arrival and so will need to choose an advisor, unless you want to risk having one chosen for you (a bad idea, generally). 

However, taking a deep dive into some narrow subfield of math doesn't limit your choices for ever. You will need to focus until you earn the degree, but then the view can open again for you if you want. 

It is probably ok to be a bit undecided as long as you aren't chaotic random. 



Note that this is a US specific answer. It wouldn't be the same in some other places.