In a series of previous papers of the first and third authors, caustics,
cut-loci, spheres and wave fronts of a system of subriemannian geodesics
issuing from a point $q_{0}$ have been studied. It happens that, outside $%
q_{0},$ only certain special arrangements of classical Lagrangian and
Legendrian singularities occur.\ As a\ consequence of this, for instance,
the generic caustic is a globally stable object, outside the origin $q_{0}.$

Here, we solve two remaining stability problems.

The first part of the paper shows that, in fact, generic caustics have
moduli at the origin, and the first module that occurs has a simple
geometric interpretation.

On the contrary, the second part of the paper shows a stability result at $q_{0}$.
We define the ''big wave front'' : it is the graph of the
multivalued function $arclength\rightarrow wave-front$, reparametrized in a
certain way. This object is a three dimensional surface, which has also the
natural structure of a wave front. The projection on the 3-dimensional space
of the singular set of this ''big wave front'' is nothing but the caustic.
We show that, in fact, this big wave front is Legendre-stable at the origin.