The celebrated positive mass theorem (PMT) of Schoen--Yau and Witten provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature, a result with far-reaching implications in geometric analysis and general relativity. I will discuss recent work to study such lower bounds in the case that black holes and/or "zero area singularities" are present in the data, restricting to the class of conformally flat manifolds.