We study quantum Hall states on curved surfaces with the aim of exposing the gravitational anomaly. We develop two general methods for computing correlation functions of the fractional quantum Hall effect (FQHE) on curved surfaces - the Ward Identity and Field Theory. We then show that on surfaces with conical singularities, the electronic fluid near the tip of the cone has an intrinsic angular momentum due solely to the gravitational anomaly. This is effect occurs because quantum Hall states behave as conformal primaries near singular points, with a conformal dimension equal to the angular momentum. We argue that the gravitational anomaly and conformal dimension determine the fine structure of the electronic density at the conical point. The singularities emerge as quasi-particles with spin and exchange statistics arising from adiabatically braiding conical singularities. Thus, the gravitational anomaly, which appears as a finite size correction on smooth surfaces, dominates geometric transport on singular surfaces.