  Moser proved in 1965 in his seminal paper that two volume forms on a compact
manifold can be conjugated by a diffeomorphism, that is to say they are
equivalent, if and only if their associated cohomology classes in the top
cohomology group of a manifold coincide. In particular, this yields a
classification of compact symplectic surfaces in terms of De Rham cohomology.
In this paper we generalize these results for volume forms admitting
transversal zeroes. In this case there is also a cohomology capturing the
classification: the relative cohomology with respect to the set of critical
hypersurface. We compare this classification scheme with the classification of
Poisson structures on surfaces which are symplectic away from a hypersurface
where they fulfill a transversality assumption ($b$-Poisson structures). We do
this using the desingularization technique introduced by
Guillemin-Miranda-Weitsman and extend it to $b^m$-Nambu structures.
