Modal phase singularities are identified in linear photonic crystals and the vortex state is explored in detail. Using group theory and phasor geometry, in the vanishing contrast limit, the modal symmetry requirements for the existence of phase singularities are determined. The vortex states are the partner functions of the symmetry groups, and hence one has a qualitative map of these modes in reciprocal space. We find that modes of even rotational symmetry are unable to form vortex states, while modes of odd rotational symmetry may form vortex states. The latter can be further classified into symmetry and accidental vortices. The insights gained using the vanishing contrast approximation are augmented by numerically solving the Maxwell's equations for the high dielectric lattice forms using the Finite Element method; the general symmetry constraints are confirmed. In addition, symmetry vortices are found to demonstrate form and locational stability over large changes in dielectric contrast, whereas this is not so for the accidental vortices, which are more sensitive to such changes.