Let f be a C-1-regular map of a closed C-infinity manifold M and Lambda be a locally maximal closed invariant set of f. We show that f vertical bar Lambda satisfies the C-1-stable specification property if and only if Lambda is a hyperbolic elementary set. We also prove that there exists a residual subset R in the space of C-1-regular maps endowed with the C-1-topology such that for f is an element of R, f vertical bar Lambda satisfies the specification property if and only if Lambda is a hyperbolic elementary set.