$W^{2,1}$ estimates for the Monge-Ampere equation $\det D^2u = f$ in $R^n$ were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to $W^{2,1}$ regularity in dimension $n \geq 3$ that have a Lipschitz singularity. 

In contrast, if $n=2$ a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are $C^1$. We will discuss a counterexample to $W^{2,1}$ regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.