A while back I declared myself a "nominalist" when it comes to linguistic theory, as opposed to a realist.
Well, hold on a minute. It's not entirely clear what I meant, because it's not entirely clear what the terms mean. In the philosophy of mathematics, "nominalism" generally means the stance that "there's no such thing as an abstract thing." [E.g. the Oxford Handbook of Philosophy of Mathematics and Logic] So on this view, not even a number really "exists," whatever that means. I think it is a little extreme to claim that there are no abstract things in this world. I'm not sure why I think that, but I do.
In the philosophy of linguistics (one of the most starved for literature), "nominalism" would have to mean something else. But it's complicated to figure out exactly what. I'll take a stab by saying that in linguistics, nominalism is a stance that says "elements of linguistic theory are not necessarily literally elements of the cognitive structure of language, although they might be."
This is quite a bit weaker than mathematical nominalism, whatever it all means.