The Eisenbud-Goto regularity conjecture says that the Castelnuovo-Mumford regularity of an embedded projective variety is bounded above by degree - codimension +1, but McCullough-Peeva recently constructed highly singular counterexamples to the conjecture. It is natural to make a precise distinction between mildly singular varieties satisfying the regularity conjecture and highly singular varieties not satisfying the regularity conjecture. In this talk, we consider the threefold case. We prove that every projective threefold with rational singularities has a nice regularity bound, which is slightly weaker than the conjectured bound, and we show that every normal projective threefold with Cohen-Macaulay Du Bois singularities in codimension two satisfies the regularity conjecture. The codimension two case is particularly interesting because one of the counterexamples to the regularity conjecture appears in this case. This is joint work with Wenbo Niu.