I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could replace variety here with scheme or whatever but I'd like things to be well-behaved enough that nonsingular has a sensible definition; I don't know exactly how well-behaved that is.

For example, given a singular variety $X$, we might ask whether $X$ falls into some natural family of objects admitting a moduli space $\mathcal{M}$ such that an open dense subset of the $K$-points of $\mathcal{M}$ correspond to nonsingular objects. Of course, when the question is formulated like this, both answers may be correct: there might also be a nonsingular variety $Y$, even quite closely related to $X$, such that $Y$ is part of a different natural family almost all of whose objects are singular. This is the kind of behaviour I'd like to know about. But from here a natural question is certainly: pick a Hilbert scheme $\operatorname{Hilb}_{\mathbb{P}^n}^P$. Are the subschemes of $\mathbb{P}^n$ it parametrises generically singular/nonsingular? Or are those subschemes generally so vicious that that question doesn't even make sense?

Finally, I don't really know anything about deformation theory, but it seems plausible that my question admits a rigorous statement and solution in that language. If anyone knows anything along these lines, I'd also be grateful to hear about that. For instance, is there a singular variety which one cannot perturb into a nonsingular one?