  We study the behavior of the modular class of an orientable Poisson manifold
and formulate some unimodularity criteria in the semilocal context, around a
(singular) symplectic leaf. Our results generalize some known unimodularity
criteria for regular Poisson manifolds related to the notion of the Reeb class.
In particular, we show that the unimodularity of the transverse Poisson
structure of the leaf is a necessary condition for the semilocal unimodular
property. Our main tool is an explicit formula for a bigraded decomposition of
modular vector fields of a coupling Poisson structure on a foliated manifold.
Moreover, we also exploit the notion of the modular class of a Poisson
foliation and its relationship with the Reeb class.
