Beauville, Bottacin and Fujiki have constructed a canonical holomorphic Poisson structure on the Hilbert scheme of points on any smooth complex Poisson surface. In the case of K3 and abelian surfaces, these varieties and their deformations give some of the few known examples of irreducible holomorphic symplectic manifolds. I will discuss the geometry and deformation theory of these Poisson varieties in the case where the Poisson surface is degenerate, as an instance of the more general theory of Poisson brackets satisfying a natural nondegeneracy condition called "holonomicity". The deformations turn out to be parametrized by the cohomology of certain "characteristic" symplectic leaves in the Hilbert scheme, which can be determined from the degeneracy strata f the Poisson surface. This talk is based on joint work in progress with Matviichuk and Schedler, and complements the talk of Schedler at this conference.