  In this paper we prove the strong Sard conjecture for sub-Riemannian
structures on 3-dimensional analytic manifolds. More precisely, given a totally
nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic
manifold, we investigate the size of the set of points that can be reached by
singular horizontal paths starting from a given point and prove that it has
Hausdorff dimension at most 1. In fact, provided that the lengths of the
singular curves under consideration are bounded with respect to a given
complete Riemannian metric, we demonstrate that such a set is a semianalytic
curve. As a consequence, combining our techniques with recent developments on
the regularity of sub-Riemannian minimizing geodesics, we prove that minimizing
sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of
class C 1 , and actually are analytic outside of a finite set of points.
