  We compute the group of Morita self-equivalences (the Picard group) of a
Poisson structure on an orientable surface, under the assumption that the
degeneracies of the Poisson tensor are linear. The answer involves mapping
class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms.
We also show that the Picard group of these structures coincides with the group
of outer Poisson automorphisms.
