Open loci results for commutative DG-rings

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give sufficient condition for the regular locus to be open, show that is always open. However, both of these loci are often empty: no matter how nice H0(A) is, there examples where A empty. then DG-module contains dense open set. Our results imply under mild hypothesis, eventually coconnective locally derived schemes generically Cohen-Macaulay, but even in very cases, they need not Gorenstein.