  We investigate the effects of ambipolar diffusion and the Hall effect on the
stability of weakly-ionized, magnetized planar shear flows. Employing a local
approach similar to the shearing-sheet approximation, we solve for the
evolution of linear perturbations in both streamwise-symmetric and
non-streamwise-symmetric geometries using WKB techniques and/or numerical
methods. We find that instability arises from the combination of shear and
non-ideal magnetohydrodynamic processes, and is a result of the ability of
these processes to influence the free energy path between the perturbations and
the shear. They turn what would be simple linear-in-time growth due to current
and vortex stretching from shear into exponentially-growing instabilities. Our
results aid in understanding previous work on the behaviour of weakly-ionized
accretion discs. In particular, the recent finding that the Hall effect and
ambipolar diffusion destabilize both positive and negative angular velocity
gradients acquires a natural explanation in the more general context of this
paper. We construct a simple toy model for these instabilities based upon
transformation operators (shears, rotations, and projections) that captures
both their qualitative and, in certain cases, exact quantitative behaviour.
