  Given a totally nonholonomic distribution of rank two on a three-dimensional
manifold we investigate the size of the set of points that can be reached by
singular horizontal paths starting from a same point. In this setting, the Sard
conjecture states that that set should be a subset of the so-called Martinet
surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture
holds in the case where the Martinet surface is smooth. Moreover, we address
the case of singular real-analytic Martinet surfaces and show that the result
holds true under an assumption of non-transversality of the distribution on the
singular set of the Martinet surface. Our methods rely on the control of the
divergence of vector fields generating the trace of the distribution on the
Martinet surface and some techniques of resolution of singularities.
