This paper deals with the identifiability of nonsmooth defects by boundary measurements, and the stability of their detection. We introduce and analyze a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiability of defects by two boundary measurements. As a consequence, we prove the unique identifiability for a large class of closed sets, including sets with an infinite number of connected components of positive capacity and totally disconnected sets. In order to rigorously justify numerical approximation results of defects by optimal design methods, we prove a geometric stability result of the defect identification process, without any a priori smoothness assumptions.