  Given a commutative noetherian non-positive DG-ring with bounded cohomology
which has a dualizing DG-module, we study its regular, Gorenstein and
Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be
open, and show that the Gorenstein locus is always open. However, both of these
loci are often empty: we show that no matter how nice $\mathrm{H}^0(A)$ is,
there are examples where the Gorenstein locus of $A$ is empty. We then show
that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded
cohomology which has a dualizing DG-module always contains a dense open set.
Our results imply that under mild hypothesis, eventually coconnective locally
noetherian derived schemes are generically Cohen-Macaulay, but that even in
very nice cases, they need not be generically Gorenstein.
