  We study left-invariant distances on Lie groups for which there exists a
one-parameter family of homothetic automorphisms. The main examples are Carnot
groups, in particular the Heisenberg group with the standard dilations. We are
interested in criteria implying that, locally and away from the diagonal, the
distance is Euclidean Lipschitz and, consequently, that the metric spheres are
boundaries of Lipschitz domains in the Euclidean sense. In the first part of
the paper, we consider geodesic distances. In this case, we actually prove the
regularity of the distance in the more general context of sub-Finsler manifolds
with no abnormal geodesics. Secondly, for general groups we identify an
algebraic criterium in terms of the dilating automorphisms, which for example
makes us conclude the regularity of homogeneous distances on the Heisenberg
group.In such a group, we analyze in more details the geometry of metric
spheres. We also provide examples of homogeneous groups where spheres presents
cusps.
