We study global non-axisymmetric oscillation modes and instabilities in magnetosphere–disc systems, as expected in neutron star X-ray binaries and possibly also in accreting black hole systems. Our 2D magnetosphere–disc model consists of a Keplerian disc in contact with a uniformly rotating magnetosphere with low plasma density. Two types of global overstable modes exist in such systems: the interface modes and the disc inertial–acoustic modes. We examine various physical effects and parameters that influence the properties of these oscillation modes, particularly their growth rates, including the magnetosphere field configuration, the velocity and density contrasts across the magnetosphere–disc interface, the rotation profile (with Newtonian or pseudo general relativistic potential), the sound speed and magnetic field of the disc. The interface modes are driven unstable by Rayleigh–Taylor and Kelvin–Helmholtz instabilities, but can be stabilized by the toroidal field (through magnetic tension) and disc differential rotation (through finite vorticity). General relativity increases their growth rates by modifying the disc vorticity outside the magnetosphere boundary. The interface modes may also be affected by wave absorption associated with corotation resonance in the disc. In the presence of a magnetosphere, the inertial–acoustic modes are effectively trapped at the innermost region of the relativistic disc just outside the interface. They are driven unstable by wave absorption at the corotation resonance, but can be stabilized by modest disc magnetic fields. The overstable oscillation modes studied in this paper have characteristic properties that make them possible candidates for the quasi-periodic oscillations observed in X-ray binaries.