  In this paper we obtain 32 canonical forms for 3D piecewise smooth vector
fields presenting the so called cusp-fold singularity. All these canonical
forms are topologically distinct and collect the main topological aspects of
the singularities described as kind of the tangencies involved and positions of
the sliding, escaping and crossing regions. Also, one-parameter bifurcations of
these canonical forms are presented and the topologically equivalent piecewise
smooth vector fields are obtained.
