  In this note, we investigate some regularity aspects for solutions of
degenerate complex Monge-Amp\`ere 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
K\"ahler-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 N\'eron-Severi group. This yields in particular the
continuity of K\"ahler-Einstein potentials on any irreducible Calabi-Yau
variety.
