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 $\phi\colon \mathbb A^2 \to C$. 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.