Radial oscillations resulting from axisymmetric perturbations in viscous accretion discs surrounding neutron stars in X-ray binary systems have been investigated. Within the framework of the alpha-viscosity model a series of hydrodynamic calculations demonstrates that the oscillations are global for alpha approximately 1. On the other hand, for alpha less-than-or-similar-to 0.4 the oscillations are local and are confined to the disc boundaries. If viscous stresses acting in the radial direction are included, however, it is found that the disc can be stabilized. The application of such instabilities in accretion discs, without reference to the boundary layer region between the neutron star (or magnetosphere) and the inner edge of the disc, to the phenomenology of quasi-periodic oscillations is brought into question.