  This paper is devoted to coregular submanifolds in Poisson geometry. We show
that their local Poisson saturation is an embedded Poisson submanifold, and we
give a normal form for this Poisson submanifold around the coregular
submanifold. This result recovers the normal form around Poisson transversals,
and it yields Poisson versions of some normal form/rigidity results around
constant rank submanifolds in symplectic geometry. As an application, we prove
a uniqueness result concerning coisotropic embeddings of Dirac manifolds in
Poisson manifolds. We also show how our results generalize to the setting of
coregular submanifolds in Dirac geometry.
