  The aim of this paper is to provide a discussion on current directions of
research involving typical singularities of 3D nonsmooth vector fields. A brief
survey of known results is presented. The main purpose of this work is to
describe the dynamical features of a fold-fold singularity in its most basic
form and to give a complete and detailed proof of its local structural
stability (or instability). In addition, classes of all topological types of a
fold-fold singularity are intrinsically characterized. Such proof essentially
follows firstly from some lines laid out by Colombo, Garc\'ia, Jeffrey,
Teixeira and others and secondly offers a rigorous mathematical treatment under
clear and crisp assumptions and solid arguments. One should to highlight that
the geometric-topological methods employed lead us to the completely
mathematical understanding of the dynamics around a T-singularity. This
approach lends itself to applications in generic bifurcation theory. It is
worth to say that such subject is still poorly understood in higher dimension.
