We analyse three codimension-two bifurcations occurring in nonsmooth systems,
when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in
continuous- and discrete-time) interacts with one of the discontinuity
boundaries characterising the system's dynamics. Rather than aiming at a
complete unfolding of the three cases, which would require specific assumptions
on both the class of nonsmooth system and the geometry of the involved
boundary, we concentrate on the geometric features that are common to all
scenarios.