Morse-Sard type results in sub-Riemannian geometry

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Measure contraction properties of Carnot groups (2015, background): “It is not even known whether general Carnot groups have negligible cut loci (this is related with the Sard conjecture in sub-Riemannian geometry [19, 11]).”
The turnpike property in finite-dimensional nonlinear optimal control (2014, background): “As it is by now well known, such an assumption is automatically satisﬁed under generic assumptions (see [16, 41]), or under controllability assumptions (see [7, 48]).”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “Because the property of being strictly normal is an open condition on geodesics (see [10, 49] and [3, Chapter: Regularity of SR distance]), after possibly shrinking U and U ′ we have that all of the minimal geodesics from x to points in U ′ are also strictly normal.”
Any sub-Riemannian metric has points of smoothness (2008, background): “This fact is proved in [7]; the proof is simple and we present it here.”
Sard Property for the endpoint map on some Carnot groups (2015, unknown): “…u is a critical point with the extra property that γ u is length minimizing for a ﬁxed sub-Riemannian structure. very general result [3] by Agrachev based on techniques of Rifford and Trélat [18] states that this set is contained in a closed nowhere dense set, for general sub-Riemannian manifolds.”