This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by “standard” inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given.