The critical state stability is investigated taking into account the plastic yield of hard and composite superconductors. The conditions have been found at which the magnetic flux jumps occur, co-developing with the plastic deformation jerks. It is shown that the interaction of mechanical and thermomagnetic instabilities can reduce appreciably the critical state stability threshold. The effect of the transport current on stability is investigated. The conditions of weak external cooling (in particular, the adiabatic case) are considered. A possible explanation of the training phenomenon is suggested.