Hörmander's propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present paper, building upon a visionary conference paper by R. Melrose \cite{Mel86}, we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremal lifts of singular curves, which are well-known curves in optimal control theory. We first revisit in depth the ideas sketched by R. Melrose in \cite{Mel86}, notably providing a full proof of its main statement. Making more explicit computations, we then explain how sub-Riemannian geometry and abnormal extremals come into play. This result shows that abnormal extremals have an important role in the classical-quantum correspondence between sub-Riemannian geometry and subelliptic operators. As a consequence, for $x\neq y$ and denoting by $K_G$ the wave kernel, we obtain that the singular support of the distribution $t\mapsto K_G(t,x,y)$ is included in the set of lengths of the normal geodesics joining $x$ and $y$, at least up to the time equal to the minimal length of a singular curve joining $x$ and $y$.