  In this article, we introduce and study singular foliations of $b^k$-type.
These singular foliations formalize the properties of vector fields that are
tangent to order $k$ along a submanifold $W \subset M$. Our first result is a
classification of these foliations, relating them to geometric structures
defined in a formal neighborhood of the submanifold, such as jets of
distributions that are involutive up to order $k-1$.
  When $W$ is a hypersurface, singular foliations of $b^k$-type are Lie
algebroids. In this particular case, they are generalizations of the
$b^k$-tangent bundles introduced by Scott. Indeed, they are always locally
isomorphic to $b^k$-tangent bundles, but globally such an isomorphism is
obstructed by a holonomy invariant. Our second main result is a
Riemann-Hilbert-style classification of singular foliations of $b^k$-type in
terms of holonomy representations.
  In this paper, we study singular foliations of $b^k$-type from several
different perspectives. In particular: (1) We study the problem of extending a
$k$-th-order foliation to a $(k+1)$-th order foliation and prove that this is
obstructed by a characteristic class. (2) When $W$ is a hypersurface, we give a
detailed study of algebroid differential forms and extend Scott's calculation
of the cohomology. (3) We study algebroid symplectic forms in terms of the
geometric structures induced on $W$. In particular, we find that there is a
close relationship between the above obstruction class for extensions and the
symplectic variation of the symplectic foliation induced on $W$.
