  Within the framework of shallow-water magnetohydrodynamics, we investigate
the linear instability of horizontal shear flows, influenced by an aligned
magnetic field and stratification. Various classical instability results, such
as H{\o}iland's growth rate bound and Howard's semi-circle theorem, are
extended to this shallow-water system for quite general profiles. Two specific
piecewise-constant velocity profiles, the vortex sheet and the rectangular jet,
are studied analytically and asymptotically; it is found that the magnetic
field and stratification (as measured by the Froude number) are generally both
stabilising, but weak instabilities can be found at arbitrarily large Froude
number. Numerical solutions are computed for corresponding smooth velocity
profiles, the hyperbolic-tangent shear layer and the Bickley jet, for a uniform
background field. A generalisation of the long-wave asymptotic analysis of
Drazin & Howard (1962) is employed in order to understand the instability
characteristics for both profiles. For the shear layer, the mechanism
underlying the primary instability is interpreted in terms of
counter-propagating Rossby waves, thereby allowing an explication of the
stabilising effects of the magnetic field and stratification.
