There is extensive literature on the stability of parallel two-phase flow, both in the context of liquid-liquid as well as gas-liquid flow. Aimed at making this literature more transparent, this paper presents a classification scheme for the various instabilities arising in parallel two-phase flow. To achieve such a classification, the equation governing the rate of change of the kinetic energy of the disturbances is evaluated for relevant values of the physical parameters. This shows the existence of five different ways of energy transfer from the primary to the disturbed flow, which have their origin in density stratification, velocity profile curvature, viscosity stratification or shear effects. Each class is discussed on the basis of references covering the developments over the last 35 years.