  Let L --> X be a complex line bundle over a compact connected Riemann
surface. We consider the abelian vortex equations on L when the metric on the
surface has finitely many point degeneracies or conical singularities and the
line bundle has parabolic structure. These conditions appear naturally in the
study of vortex configurations with constraints, or configurations invariant
under the action of a finite group. We first show that the moduli space of
singular vortex solutions is the same as in the regular case. Then we compute
the total volume and total scalar curvature of the moduli space singular vortex
solutions. These numbers differ from the case of regular vortices by a very
natural term. Finally we exhibit explicit non-trivial vortex solutions over the
thrice punctured hyperbolic sphere.
