Poisson structures vanishing linearly on a set of smooth closed disjoint
curves are generic in the set of all Poisson structures on a
compact connected oriented surface. W construct a complete set
of invriants classifying these structures up to an
orient-preserving Poisson isomorphism. We show that there is a
set of non-trivial infinitesimal deformations which generate
the second Poisson cohomology and such that each of the deformations
changes exactly one of the classifying invarients. As an
example, we consider Poisson structures on the sphere which vanish
linearly on a set of smooth closed disjoint curves.