Not yet reviewed by Pith; the record is open.

We investigate the crystalline stationary states of a dipolar Bose-Einstein condensate in a planar trapping geometry. Our focus is on the ground state phase diagram in the thermodynamic limit, where triangular, honeycomb and stripe phases occur. We quantify the superfluid fraction by calculating the non-classical translational inertia, which allows us to identify favorable parameter regimes for observing supersolid ground states. We develop two simplified theories to approximately describe the ground states, and consider the relationship to roton softening in the uniform ground state. This also allows us to extend the phase diagram to the low density regime. While the triangular and honeycomb states have an isotropic superfluid response tensor, the stripe state exhibits anisotropic superfluidity.