  We study the collapse of ultracold fermionic gases into inhomogeneous states
due to strong dipolar interaction in both 2D and 3D. Depending on the
dimensionality, we find that two different types of inhomogeneous states are
stabilized once the dipole moment reaches a critical value $d>d_c$: the {\it
stripe phase} and {\it phase separation} between high and low densities. In 2D,
we prove that the stripe phase is always favored for $d\gtrsim d_c$, regardless
of the microscopic details of the system. In 3D, the one-loop perturbative
calculation suggests that the same type of instability leads to phase
separation. Experimental detection and finite-temperature effects are
discussed.
