  We discuss a novel instability in inertia-less electron magneto-hydrodynamics
(EMHD), which arises from a combination of electron velocity shear and electron
density gradients. The unstable modes have a lengthscale longer than the
transverse density scale, and a growth-rate of the order of the inverse Hall
timescale. We suggest that this density-shear instability may be of importance
in magnetic reconnection regions on scales smaller than the ion skin depth, and
in neutron star crusts.
  We demonstrate that the so-called Hall drift instability, previously argued
to be relevant in neutron star crusts, is a resistive tearing instability
rather than an instability of the Hall term itself. We argue that the
density-shear instability is of greater significance in neutron stars than the
tearing instability, because it generally has a faster growth-rate and is less
sensitive to geometry and boundary conditions.
  We prove that, for uniform electron density, EMHD is "at least as stable" as
regular, incompressible MHD, in the sense that any field configuration that is
stable in MHD is also stable in EMHD. We present a connection between the
density-shear instability in EMHD and the magneto-buoyancy instability in
anelastic MHD.
