  In this paper we extend the classification scheme in [S] for $b^m$-symplectic
surfaces and, more generally, $b^m$-Nambu structures to the equivariant
setting. When the compact group is the group of deck-transformations of an
orientable covering, this yields the classification of these objects for
non-orientable surfaces. The paper also includes recipes to construct
$b^m$-symplectic structures on surfaces. Feasibility of such constructions
depends on orientability and on the colorability of an associated graph. The
desingularization technique in [GMW] is revisited for surfaces and the
compatibility with this classification scheme is analyzed. We recast the
strategy used in [Mt] to classify stable Nambu structures of top degree on
orientable manifolds to classify $b^m$-Nambu structures (not necessarily
oriented) using the language of $b^m$-cohomology. The paper ends up with an
equivariant classification theorem of $b^m$-Nambu structures of top degree.
