  In this paper we consider surfaces of class $C^1$ with continuous prescribed
mean curvature in a three-dimensional contact sub-Riemannian manifold and prove
that their characteristic curves are of class $C^2$. This regularity result
also holds for critical points of the sub-Riemannian perimeter under a volume
constraint. All results are valid in the first Heisenberg group $\mathbb{H}^1$.
