The linear stability of thin vertically isothermal density-stratified Keplerian discs in toroidally dominated magnetic fields is treated by asymptotic expansions in the small aspect ratio of the discs. The discs are found to be spectrally stable. The great variety of possible initial conditions leads to three regimes of non-exponential growth of perturbations, which are classified according to different relative levels of the in-plane and axial perturbed velocities. The first two regimes of instability are characterized by the decoupling of the magnetosonic (MS) and inertia–Coriolis (IC) modes, as well as by algebraic temporal growth of the perturbations, which are driven by either MS or IC modes (hereafter MS and IC regimes of instability, respectively). The third, mixed IC–MS regime of non-exponential, non-algebraic growth is due only to non-axisymmetric perturbations. The latter regime is characterized by high radial and azimuthal wavenumbers, and growth time of the order of tens of rotating periods. The mixed IC–MS regime most likely exhibits the maximal growth as compared with the IC and MS regimes. In the first two regimes of instability the compressible MS mode plays a principal role either as the driver of the growth or the driven growing mode, while the mixed IC–MS regime is described by the Boussinesq approximation for incompressible fluid. The latter is obtained as a natural limit of the expansion scheme. The presence of magnetic field in the mixed IC–MS regime may drastically increase the growth rates of the perturbations as compared with the pure hydrodynamic system.