A useful (at least for me) example is given in Kollar's article/book on resolutions of singularities about how you can't expect to get a "resolution functor": take a quadric cone $$C = {(x,y,z) in mathbb A^3: xy-z^2=0}$$ in $mathbb A^3$. Then you have the obvious map $phicolon mathbb A^2 to C$. But now suppose that $C'$ is a resolution of $C$ provided by a putative "resolution functor". Then if we let $tilde{C}$ be the minimal resolution, $C'$ factors through $C$. If we assume that $mathbb A^2$ is resolved by itself (as seems reasonable!) then we'd have to have $phi$ lifting to a map $mathbb A^2 to tilde{C}$ compatibly with the original morphism, which of course one cannot do.
I found the introduction to Kollar's article really useful in understanding what one can and cannot expect from resolution of singularities.
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