  Recent breakthrough experiments on dipolar condensates have reported the
creation of supersolids, including two-dimensional arrays of quantum droplets.
Droplet arrays are, however, not the only possible non-trivial density
arrangement resulting from the interplay of mean-field instability and quantum
stabilization. Several other possible density patterns may occur in trapped
condensates at higher densities, including the so-called honeycomb supersolid,
a phase that exists, as it is also the case of a triangular droplet supersolid,
in the thermodynamic limit. We show that compared to droplet supersolids,
honeycomb supersolids have a much-enhanced superfluid fraction while keeping a
large density contrast, and constitute in this sense a much better dipolar
supersolid. However, in contrast to droplet supersolids, quantized vortices
cannot be created in a honeycomb supersolid without driving a transition into a
so-called labyrinthic phase. We show that the reduced moment of inertia, and
with it the superfluid fraction, can be however reliably probed by studying the
dynamics following a scissors-like perturbation.
