In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following:
The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: $(X,p) \sim (Y,q)$ if there exists a pointed scheme $(Z,r)$ admitting smooth morphisms $(Z,r) \to (X,p)$ and $(Z,r) \to (Y,q)$.
This relation is obviously reflexive and symmetric; to check transitivity, use the fact that pullbacks of smooth morphisms are smooth.
On its surface, this appears, at least to me, to have a different flavor from other ways in which singularities are studied. (I'm thinking in particular of the study of deformations of singularities, but I know very little about this subject, so my comment should not be taken too seriously.) Thus, I'd like to know what is known about the "singularity types" of this definition, other than the fact that all of them show up in each of a large number of moduli spaces.
I'm really interested in getting a general "flavor" of the theory, but since this is a question-and-answer site, here's a specific question:
Can infinitely many singularity types occur on a single scheme $X$ that is of finite type over a field?
Obviously, infinitely many singularity types show up in the moduli spaces for which Vakil proves Murphy's Law. But if I understand correctly, these spaces are only locally noetherian, so it is still possible that only finitely many singularity types show up on each connected component.