  We study global non-axisymmetric oscillation modes trapped near the inner
boundary of an accretion disc. Observations indicate that some of the
quasi-periodic oscillations (QPOs) observed in the luminosities of accreting
compact objects (neutron stars, black holes and white dwarfs) are produced in
the inner-most regions of accretion discs or boundary layers. Two simple models
are considered in this paper: The magnetosphere-disc model consists of a thin
Keplerian disc in contact with a uniformly rotating magnetosphere with and low
plasma density, while the star-disc model involves a Keplerian disc terminated
at the stellar atomosphere with high density and small density scale height. We
find that the interface modes at the magnetosphere-disc boundary are generally
unstable due to Rayleigh-Taylor and/or Kelvin-Helmholtz instabilities. However,
differential rotation of the disc tends to suppress Rayleigh-Taylor instability
and a sufficiently high disc sound speed (or temperature) is needed to overcome
this suppression and to attain net mode growth. On the other hand,
Kelvin-Helmholtz instability may be active at low disc sound speeds. We also
find that the interface modes trapped at the boundary between a thin disc and
an unmagnetized star do not suffer Rayleigh-Taylor or Kelvin-Helmholtz
instability, but can become unstable due to wave leakage to large disc radii
and, for sufficiently steep disc density distributions, due to wave absorption
at the corotation resonance in the disc. The non-axisymmetric interface modes
studied in this paper may be relevant to the high-frequency QPOs observed in
some X-ray binaries and in cataclysmic variables.
