  Several types of generically-nondegenerate Poisson structures can be
effectively studied as symplectic structures on naturally associated Lie
algebroids. Relevant examples of this phenomenon include log-, elliptic,
$b^k$-, scattering and elliptic-log Poisson structures. In this paper we
discuss topological obstructions to the existence of such Poisson structures,
obtained through the characteristic classes of their associated symplectic Lie
algebroids. In particular we obtain the full obstructions for surfaces to carry
such Poisson structures.
