  We determine the relative stability of different ground-state phases of
spin-imbalanced popula- tions of attractive fermions in square lattices. The
phases are systematically characterized by the symmetry of the order parameter
and the real- and momentum-space structures using Hartree- Fock-Bogoliubov
theory. We find several type of unidirectional Larkin-Ovchinikov-type phases.
We discuss the effect of commensuration between the ordering wave vector and
the density imbalance, and describe the mechanism of Fermi surface
reconstruction and pairing for various orders. A robust supersolid phase is
shown to exist when the ordering wave vector is diagonally directed.
