A sub-Riemannian manifold is a manifold $M$ endowed with a distinguished subbundle $HM$ of the tangent bundle $TM$ and with a metric on $HM$. A distance on $M$ can be defined on minimizing the length among curves which are tangent to $HM$. One of the main open problems in the field is the regularity of length minimizers: this is not trivial due to the presence of the so called abnormal curves. We provide a characterization of abnormal curves in stratified Lie groups showing that these curves are contained in certain algebraic varieties; applications to the problem of geodesics' regularity will be discussed. This is based on a joint work with E. Le Donne, G. P.
Leonardi and R. Monti.