  We study the density instabilities of a two-dimensional gas of dipolar
fermions with aligned dipole moments. We show that the Random Phase
Approximation (RPA) for the density-density response function is never accurate
for the dipolar gas. We incorporate correlations beyond RPA via an improved
version of the Singwi-Tosi-Land-Sjolander scheme. In addition to density-wave
instabilities, our formalism captures the collapse instability that is expected
from Hartree-Fock calculations but is absent from RPA. Crucially, we find that
when the dipoles are perpendicular to the layer, the system spontaneously
breaks rotational symmetry and forms a stripe phase, in defiance of
conventional wisdom.
