This work is concerned with conditions ensuring stability of a given solution of asystem of nonlinear equations with respect to large (not asymptotically thin) classesof right-hand side perturbations. Our main focus is on those solutions that are in a sensesingular, and hence, their stability properties are not guaranteed by “standard” inversefunction-type theorems. In the twice differentiable case, these issues have receivedsome attention in the existing literature. Moreover, a few results in this direction areknown in the case when the first derivative is merely B-differentiable. Here, we furtherelaborate on a similar setting, but the main attention is paid to the case of piecewisesmooth equations. Specifically, we study the effect of singularity of a solution for someactive smooth selection on the overall stability properties, and we provide sufficientconditions ensuring the needed stability properties in the cases when such smoothselections may exist. Finally, an application to a piecewise smooth reformulation ofcomplementarity problems is given.