  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.
