  Protoplanetary discs should exhibit a weak vertical variation in their
rotation profiles. Typically this `vertical shear' issues from a baroclinic
effect driven by the central star's radiation field, but it might also arise
during the launching of a magnetocentrifugal wind. As a consequence,
protoplanetary discs are subject to a hydrodynamical instability, the `vertical
shear instability' (VSI), whose breakdown into turbulence could transport a
moderate amount of angular momentum and facilitate, or interfere with, the
process of planet formation. Magnetic fields may suppress the VSI, however,
either directly via magnetic tension or indirectly through magnetorotational
turbulence. On the other hand, protoplanetary discs exhibit notoriously low
ionisation fractions, and non-ideal effects, if sufficiently dominant, may come
to the VSI's rescue. In this paper we develop a local linear theory that
explores how non-ideal MHD influences the VSI, while also launching additional
diffusive shear instabilities. We derive a set of analytical criteria that
establish when the VSI prevails, and then show how it can be applied to a
realistic global model of a protoplanetary disc. Our calculations suggest that
within ~10au the VSI should have little trouble emerging in the main body of
the disk, but beyond that, and in the upper regions of the disc, its onset
depends sensitively on the size of the preponderant dust grains.
