This paper explores special bifurcations that appear in dynamical systems possessing codimension-2 discontinuity sets in the state space. In these systems, the vector field has continuously many directional limits at the discontinuity set. It is shown that the trajectories can reach the discontinuity set along specific limit directions. The number and type of these organising objects characterise the behaviour of the dynamics in the vicinity of the discontinuity set. Thus, bifurcations of limit directions have a remarkable effect on the system. In the paper, two special bifurcations are explored; the tangency bifurcation and the saddle–node bifurcation of limit direction are followed through different formulations of the dynamical system. It is demonstrated that these bifurcations appear in rigid body dynamics in the presence of dry friction.