In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘two-fold’ singularities are difficult to find amongst physical models. They occur where the relative curvature between the flow field and the switching manifold is nonsymmetric about the discontinuity. Here we motivate their study with a local form model of nonlinear control that exhibits the two-fold singularity, where the flow is constant either side of a curved switching manifold. We describe the local dynamics around general two-fold singularities, then consider their effect on global dynamics via one parameter bifurcations of limit cycles.