One knows, of course, that Hilbert schemes of smooth curves and smooth surfaces are nice, and Hilbert schemes of varieties of dimension greater than two are terrifying.
Zhiwei Yun was here giving a talk about his work with Davesh Maulik on Hilbert schemes of curves with planar singularities, and he made a point I’d never appreciated; it’s not the dimension of the variety, but the dimension of its tangent space that really measures the terrifyingness of the Hilbert space. Singular curves C with planar singularities are not so bad — you still have a nice Hilbert scheme with an Abel-Jacobi map to the compactified Jacobian. But let C be the union of the coordinate axes in A^3 and all bets are off. Hideous extra high-dimensional components aplenty. If I had time to write a longer blog post today I would think about what the punctual Hilbert scheme at the origin looks like. But maybe one of you guys will just tell me.
Update: Jesse Kass explains that I am wrong about C; its Hilbert scheme has a non-smoothable component, but it doesn’t have any components whose dimension is too large.