The oscillation spectrum of a spatially inhomogeneous current-carrying plasma is analysed in conditions where all the plasma electrons are in motion relative to quiescent ions. Along with the density and temperature of the particles, the directed motion of the electrons, i.e. the plasma current, is considered to be inhomogeneous. The analysis shows that in certain cases this directed motion can be a cause of instability in a current-carrying plasma. The author establishes criteria for the instabilities of such a plasma in various frequency ranges and suggests ways of averting them. In particular, he draws attention to a dynamic method of suppressing instabilities by means of the current itself, in conditions where the plasma current is variable in time.