We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient are rotation-free respectively divergence-free components of the vector field. The sinks and sources respectively vortices are then automatically identified as the critical points of the corresponding scalar-valued potentials. The global nature of the decomposition avoids the approximation problem of the Jacobian and higher order tensors used in local methods, while the two potentials plus a harmonic flow component are an exact decomposition of the vector field containing all information.