  A $b$-contact structure on a $b$-manifold $(M,Z)$ is a singular Jacobi
structure on $M$ satisfying a transversality condition along the hypersurface
$Z$. We show that, in three dimensions, $b$-contact structures with overtwisted
three-dimensional leaves satisfy an existence $h$-principle that allows
prescribing the induced singular foliation. We give a method to classify
$b$-contact structures on a given $b$-manifold and use it to give a
classification on $S^3$ with either a two-sphere or an unknotted torus as the
critical surface. We also discuss generalizations to higher dimensions.
