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Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. By M-ultrafilter, I mean that U measures the subsets of x which are in M.

My question is, by varying U, how much can I affect the ultrapower of M by U? Let's say I limit myself to a U which are countably complete, so that the ultrapower will be wellfounded. If this question is too vague or broad, I'd welcome any interesting examples of things that are possible or impossible.

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As much as you wish. Lowenheim Skolem give you such situations and then you can affect it too much. For instance, you can construct situations where the critical point is singular in the ultrapower. You cannot do much if U is amenable to M. Then it is like a real ultrafilter. I don't know what you are asking actually.

An interesting question is whether you can have an ultrafilter on kappa such that the powerset of kappa^+ is in the ultrapower and James Cummings solved this by showing that you can. I don't know if it is interesting to look for ultrafilters that can code powerset(kappa^++) into the ultrapower. probably his proof already gives that.

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