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

The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces