In this note, we investigate some regularity aspects for solutions of degenerate complex Monge-Ampère equations (DCMAE) on singular spaces. First, we study the Dirichlet problem for DCMAE on singular Stein spaces, showing a general continuity result. A consequence of our results is that Kähler-Einstein potentials are continuous at isolated singularities. Next, we establish the global continuity of solutions to DCMAE when the reference class belongs to the real Néron-Severi group. This yields in particular the continuity of Kähler-Einstein potentials on any irreducible Calabi-Yau variety. We show that the associated singular Kähler-Einstein metrics are strictly positive when the variety is moreover smoothable.