It's vague. Is there a specific reaction or set of reactions that you're interested in, in a particular kind of situation/environment? Otherwise, I don't really know what to say. There are some with oxygen and some without. I have no idea what "so many" means here.
In most gen chem texts (or web search) there is a table or list of common polyatomic ions. It seems the vast majority of them contain oxygen. A few exceptions are CN-, and NH4+. I am curious if there is a physical reason for this pattern.
"Reason" is that molecules with oxygen are typically quite stable with a negative charge. Especially if the molecule contains a resonance structure near the oxygen. Since oxygen is very electronegative it doesn't mind keeping the charge to other molecules as easily as those without oxygen. Could be more technical I guess but that's the gist.
Nice question. I agree with ChaseRLewis. By definition an anion is a negatively charged ion that easily combines with metal cations or protons to form a salt or an acid. In order for an ion to be capable of easily donating electrons it must have a very polar or easily polarisable structure which at the same time must be stable. The most electronegative elements are fluorine and oxygen so they should form the most polar bonds, but flourine does not form stable dianions. There are, however some oxygen-free ions with relatively covalent bonds, which are relatively stable, such as (HS)-, (CN)- , (SCN)-, NH2- and a few others such as (N3)- (C2)2- or (RC2)2- whose compounds with metals are generally very unstable.