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In this paper we prove the strong Sard conjecture for sub-Riemannianstructures on 3-dimensional analytic manifolds. More precisely, given a totallynonholonomic analytic distribution of rank 2 on a 3-dimensional analyticmanifold, we investigate the size of the set of points that can be reached bysingular horizontal paths starting from a given point and prove that it hasHausdorff dimension at most 1. In fact, provided that the lengths of thesingular curves under consideration are bounded with respect to a givencomplete Riemannian metric, we demonstrate that such a set is a semianalyticcurve. As a consequence, combining our techniques with recent developments onthe regularity of sub-Riemannian minimizing geodesics, we prove that minimizingsub-Riemannian geodesics in 3-dimensional analytic manifolds are always ofclass C 1 , and actually are analytic outside of a finite set of points.

Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3