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In this article, we introduce and study singular foliations of $b^k$-type.These singular foliations formalize the properties of vector fields that aretangent to order $k$ along a submanifold $W \subset M$. Our first result is aclassification of these foliations, relating them to geometric structuresdefined in a formal neighborhood of the submanifold, such as jets ofdistributions that are involutive up to order $k-1$.  When $W$ is a hypersurface, singular foliations of $b^k$-type are Liealgebroids. In this particular case, they are generalizations of the$b^k$-tangent bundles introduced by Scott. Indeed, they are always locallyisomorphic to $b^k$-tangent bundles, but globally such an isomorphism isobstructed by a holonomy invariant. Our second main result is aRiemann-Hilbert-style classification of singular foliations of $b^k$-type interms of holonomy representations.  In this paper, we study singular foliations of $b^k$-type from severaldifferent 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 isobstructed by a characteristic class. (2) When $W$ is a hypersurface, we give adetailed study of algebroid differential forms and extend Scott's calculationof the cohomology. (3) We study algebroid symplectic forms in terms of thegeometric structures induced on $W$. In particular, we find that there is aclose relationship between the above obstruction class for extensions and thesymplectic variation of the symplectic foliation induced on $W$.

Jets of foliations and $b^k$-algebroids