We construct examples of solutions to Monge-Ampère equations with point masses whose singular sets exhibit polyhedral structures. We also construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on lower-dimensional sets, and study the regularity of such solutions away from these singular sets. To motivate these constructions, we present examples arising from optimal transport where the potentials of the optimal transport maps satisfy Monge-Ampère equations similar to the ones we study. In particular, we focus on planar optimal transport problems where the target domain is nonconvex.