We study singular 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 bm-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. In the last section of this paper we tackle the problem of existence of bm-contact structures on a given manifold. We prove that convex hypersurfaces can be realized as critical set of b2k-contact structures. In particular, in the 3-dimensional case, this construction yields the existence of a generic set of surfaces Z such that the pair (M;Z) is a b2k-contact manifold and Z is its critical hypersurface