When is a blow-up non-singular?

There is no general criterion, as far as I know, it is all try and see.

Any projective birational morphism $f:X\to Y$ between varieties is the blowup of some sheaf of ideals $I$ on $Y$, so you can see that anything can happen.

Craig Huneke told me about this paper: "On the smoothness of blow-ups" (MR1446135, by O'Carroll and Valla). The title alone seems to suggest it might be useful for you.

Although there is a good reason that $(x,y)^2$ has a smooth blow-up. It is a power of an ideal which itself has a smooth blowup. See for example Hartshorne, Algebraic Geometry, Chapter II, Section 7, Exercise 7.11.

I suspect that, on smooth surfaces, one can probably say more, via "Zariski-factorization" type ideas, but I'm not sure what the right answer would be.

Edit: I've looked around for a good reference on "Zariski-Factorization", but I'm not sure what a good one is. Does someone know?