The regularity of length-minimizing curves on sub-Riemannian manifolds is an open problem, of which quite little is known compared to the Riemannian setting, where classically all minimizers are smooth. The main issue in the sub-Riemannian case is the existence of abnormal minimizers which do not satisfy any geodesic equation. In this talk I will give an overview of the problem and describe a series of results obtained by studying the tangent cone of a minimizing curve. The benefit of the approach is that it allows simplifying the problem to studying minimizers in sub-Riemannian Carnot groups. The group structure is then used for a constructive linear algebraic method to shorten curves, which can prove the non-minimality of some class of curves including for instance curves with corner type singularities. This talk is based on joint work with Enrico Le Donne.