We present the first examples of nonsmooth sub-Riemannian length minimizing curves. The length minimizer with the lowest regularity within these examples is of class $C^2\setminus C^3$. The singularity is at a boundary point. The result is sharp in the sense that we can prove that, within these examples, it is not possible to find a minimizer of class $C^1\setminus C^2$. This is a joint work with Y. Chitour, F. Jean, R. Monti, L. Rifford, L. Sacchelli, and M. Sigalotti.