Given a lie group $G$ that acts over a symplectic manifold $(M,\omega)$ then its quotient $M/G$ (under mild hipothesis) is a Poisson manifold.
This is an very useful way to construct Poisson manifolds from symplectic manifolds alreadly known.
I am asking myself if there is an example of a Poisson manifold that do not arises as a quotient of symplectic manifold. Maybe a fully degenerated Poisson bracket can gives us some nice examples?