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We undertake a detailed study of the geometry of Bottacin's Poisson structures on Hilbert schemes of points in Poisson surfaces, i.e. smooth complex surfaces equipped with an effective anticanonical divisor. We focus on three themes that, while logically independent, are linked by the interplay between (characteristic) symplectic leaves and deformation theory. Firstly, we construct the symplectic groupoids of the Hilbert schemes and develop the classification of their symplectic leaves, using the methods of derived symplectic geometry. Secondly, we establish local normal forms for the Poisson brackets, and combine them with a toric degeneration argument to verify that Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson manifolds in terms of the geometry of the modular vector field. Finally, using constructible sheaf methods, we compute the space of first-order Poisson deformations when the anti-canonical divisor is reduced and has only quasi-homogeneous singularities. (The latter is automatic if the surface is projective.) Along the way, we find a tight connection between the Poisson geometry of the Hilbert schemes and the finite-dimensional Lie algebras of affine transformations, which is mediated by syzygies. In particular, we find that the Hilbert scheme has a natural subvariety that serves as a global counterpart of the nilpotent cone, and we prove that the Lie algebras of affine transformations have holonomic dual spaces -- the first such series of Lie algebras to be discovered.