  We characterize the absolute continuity of the law and the Malliavin-Sobolev
regularity of random nodal volumes associated with smooth Gaussian fields on
generic $\mathcal{C}^2$ manifolds with arbitrary dimension. Our results extend
and generalize the seminal contribution by Angst and Poly (2020) about
stationary fields on Euclidean spaces and cover, in particular, the case of
two-dimensional manifolds, possibly with boundary and corners. The main tools
exploited in the proofs include the use of Gaussian measures on Banach spaces,
Morse theory, and the characterization of Malliavin-Sobolev spaces in terms of
ray absolute continuity. Several examples are analyzed in detail.
