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.