  We study the absolute continuity with respect to the Lebesgue measure of the
distribution of the nodal volume associated with a smooth, non-degenerated and
stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions,
we prove that in dimension $d\geq 3$, the distribution of the nodal volume has
an absolutely continuous component plus a possible singular part. This singular
part is actually unavoidable baring in mind that some Gaussian processes have a
positive probability to keep a constant sign on some compact domain. Our
strategy mainly consists in proving closed Kac--Rice type formulas allowing one
to express the volume of the set $\{f =0\}$ as integrals of explicit
functionals of $(f,\nabla f,\text{Hess}(f))$ and next to deduce that the random
nodal volume belongs to the domain of a suitable Malliavin gradient. The
celebrated Bouleau--Hirsch criterion then gives conditions ensuring the
absolute continuity.
