Beam instabilities in the strongly magnetized electron–positron plasma of a pulsar magnetosphere are considered. We analyse the resonance conditions and estimate the growth rates of the Cherenkov and cyclotron instabilities of the ordinary (O), extraordinary (X) and Alfvén modes in two limiting regimes: kinetic and hydrodynamic. The importance of the different instabilities as a source of coherent pulsar radiation generation is then estimated, taking into account the angular dependence of the growth rates and the limitations on the length of the coherent wave–particle interaction imposed by the curvature of the magnetic field lines. We conclude that in the pulsar magnetosphere, Cherenkov-type instabilities occur in the hydrodynamic regime, while cyclotron-type instabilities occur in the kinetic regime. We argue that electromagnetic cyclotron-type instabilities on the X, O and probably Alfvén waves are more likely to develop in the pulsar magnetosphere.