  Let $Z$ be a hypersurface of a manifold $M$. The $b$-tangent bundle of $(M,
Z)$, whose sections are vector fields tangent to $Z$, is used to study
pseudodifferential operators and stable Poisson structures on $M$. In this
paper we introduce the $b^k$-tangent bundle, whose sections are vector fields
with "order $k$ tangency" to $Z$. We describe the geometry of this bundle and
its dual, generalize the celebrated Mazzeo-Melrose theorem of the de Rham
theory of $b$-manifolds, and apply these tools to classify certain Poisson
structures on compact oriented surfaces.
