  In this article, we study the Dirac spectrum of typical hyperbolic surfaces
of finite area, equipped with a nontrivial spin structure (so that the Dirac
spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$
with $o(\sqrt{g})$ cusps, we prove convergence of the spectral density to the
spectral density of the hyperbolic plane, with quantitative error estimates.
This result implies upper bounds on spectral counting functions and
multiplicities, as well as a uniform Weyl law, true for typical hyperbolic
surfaces equipped with any nontrivial spin structure.
