In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure. In particular, given an almost contact manifold $M$ with a hypersurface $Z$, this yields the existence of a $b^{2k}$-contact structure on $M$ realizing $Z$ as a critical set. As a consequence of the desingularization techniques in [GMW], we prove the existence of folded contact forms on any almost contact manifold.

28 pages, new high dimensional construction added