The three-dimensional linear stability of the two-dimensional, incompressible flow is studied numerically in plane channels, exhibiting sudden expansions/constrictions in the form of steps. The corner singularities of the geometries lead to steep gradients in the flow quantities in the vicinity of the steps, where the flow separates immediately, even at low Reynolds numbers. The geometries of the systems considered, i.e. the backward-facing-step, forward-facing-step and plane sudden-expansion problems, are varied in a systematic way such that a wide range of the parameter space is covered. A global, temporal linear stability analysis shows that the resulting stability boundaries are continuous functions of the geometry parameters. If the critical Reynolds and wave numbers are scaled appropriately, they approach a linear asymptotic behaviour in the limit of very large as well as small step heights. An energy-transfer analysis is used to understand the physical nature of the instabilities. This analysis reveals that the basic flow becomes unstable due to different instability mechanisms, which are studied in detail for representative geometries.