  We explore the linear stability of astrophysical discs exhibiting vertical
shear, which arises when there is a radial variation in the temperature or
entropy. Such discs are subject to a "vertical-shear instability", which recent
nonlinear simulations have shown to drive hydrodynamic activity in the
MRI-stable regions of protoplanetary discs. We first revisit locally isothermal
discs using the quasi-global reduced model derived by Nelson et al. (2013).
This analysis is then extended to global axisymmetric perturbations in a
cylindrical domain. We also derive and study a reduced model describing discs
with power law radial entropy profiles ("locally polytropic discs"), which are
somewhat more realistic in that they possess physical (as opposed to numerical)
surfaces. In all cases the fastest growing modes have very short wavelengths
and are localised at the disc surfaces (if present), where the vertical shear
is maximal. An additional class of modestly growing vertically global body
modes is excited, corresponding to destabilised classical inertial waves
("r-modes"). We discuss the properties of both types of modes, and stress that
those that grow fastest occur on the shortest available length scales
(determined either by the numerical grid or the physical viscous length). This
ill-posedness makes simulations of the instability difficult to interpret. We
end with some brief speculation on the nonlinear saturation and resulting
angular momentum transport.
