We investigate existence, stability, and instability of anchored rotating spiral waves in a model for geometric curve evolution. We find existence in a parameter regime limiting on a purely eikonal curve evolution. We study stability and instability both theoretically in this limiting regime and numerically, finding both oscillatory, at first convective instability, and saddle-node bifurcations. Our results in particular shed light onto instability of spiral waves in reaction-diffusion systems caused by an instability of wave trains against transverse modulations.