  A two-fold singularity is a point on a discontinuity surface of a
piecewise-smooth vector field at which the vector field is tangent to the
surface on both sides. Due to the double tangency, forward evolution from a
two-fold is typically ambiguous. This is an especially serious issue for
two-folds that are reached by the forward orbits of a non-zero measure set of
initial points. The purpose of this paper is to explore the concept of
perturbing the vector field so that forward evolution is well-defined, and
characterising the perturbed dynamics in the limit that the size of the
perturbation tends to zero. This concept is applied to a two-fold in two
dimensions. Three forms of perturbation: hysteresis, time-delay, and noise, are
analysed individually. In each case, the limit leads to a novel probabilistic
notion of forward evolution from the two-fold.
