We study the nature of finite-time singularities for the Chern-Ricci flow, partially answering a question of Tosatti-Weinkove. We show that a solution of degenerate parabolic complex Monge-Ampère equations starting from arbitrarily positive (1,1)-currents are smooth outside some analytic subset, generalizing works by Di Nezza-Lu. We extend Guedj-Lu's recent approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Hermitian manifolds. We apply it to studying the Chern-Ricci flows on complex log terminal varieties starting from an arbitrary current.

final version, to appear in Analysis & PDE