  We look at Poisson geometry taking the viewpoint of singular foliations,
understood as suitable submodules generated by Hamiltonian vector fields rather
than partitions into (symplectic) leaves. The class of Poisson structures which
behave best from this point of view, are those whose submodule generated by
Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them
almost regular Poisson structures and determine them completely. They include
regular Poisson and log symplectic manifolds, as well as several other Poisson
structures whose symplectic foliation presents singularities.
  We show that the holonomy groupoid associated with an almost regular Poisson
structure is a Poisson groupoid, integrating a naturally associated Lie
bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as
such it provides a desingularization. The holonomy groupoid is "minimal" among
Lie groupoids which give rise to the submodule generated by Hamiltonian vector
fields. This implies that, in the case of log-symplectic manifolds, the
holonomy groupoid coincides with the symplectic groupoid constructed by
Gualtieri and Li. Last, we focus on the integrability of almost regular Poisson
manifolds and exhibit the role of the second homotopy group of the
source-fibers of the holonomy groupoid.
