  We investigate the spectrum of the spin Dirac operator on families of
hyperbolic surfaces where a set of disjoint simple geodesics shrink to $0$,
under the hypothesis that the spin structure is non-trivial along each pinched
geodesic. The main tool is a trace formula for the Dirac operator on finite
area hyperbolic surfaces. We derive a version of Huber's theorem and a
non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces
with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues
of the Dirac operator which is \emph{uniform} in the degenerating parameter.
The main result is the convergence of the Selberg zeta function associated to
the Dirac operator on such families of hyperbolic surfaces. A central role is
played by a $\{ \pm 1 \}$-valued class function $\varepsilon$ determined by the
spin structure.
