We describe the effects of both magnetic buoyancy and differential rot
ation on a disc of isothermal gas embedded in a purely azimuthal magne
tic field, in order to study the evolution and interplay of Parker and
shearing instabilities.We perform a linear analysis of the evolution
of perturbations in the shearing sheet model. Both instabilities occur
on the slow MHD branch of the dispersion relation, and can affect the
same waves. We put a stress on the natural polarization properties of
the slow MHD waves to get a better understanding of the physics invol
ved. The mechanism of the shearing instability is described in details
. Differential rotation can transiently stabilize slow MHD waves with
a vertical wavelength longer than the scale height of the disc, agains
t the Parker instability. Waves with a vertical wavelength shorter tha
n the scale height of the disc are subject to both the Parker and the
transient shearing instabilities. They occur in different ranges of ra
dial wavenumbers, i.e. at different times in the shearing evolution; t
hese ranges can overlap or, on the contrary, be separated by a phase o
f wave-like oscillations, depending on the strength of differential ro
tation. These analytical results, obtained in a WKB approximation, are
found to be in excellent agreement with numerical solutions of the fu
ll set of linearized equations. Our results can be applied to both gal
actic and accretion discs.