  The goal of this paper is to use singularity theory to find normal forms near
the critical points of the sub-Riemannian exponential map. Three cases are
studied: the $\alpha$-Grushin plane with fold singularities, and the special
unitary group $\mathrm{SU}(2)$ and special linear group $\mathrm{SL}(2)$ with
fold and saddle-like singularities. They serve as examples of different
sub-Riemannian structures and the techniques presented can be applied to other
contexts. The paper also includes a discussion of the implications of this
approach, as well as open problems.
