A thermodynamic theory of stability in solids with intrinsic dissipation of plastic type is developed, and various related constitutive inequalities are discussed. The thermodynamic formalism for finite strain elastoplasticity is presented, with rate-dependent plastic behaviour and its rate-independent limit described with the help of internal variables. A general condition sufficient for stability of equilibrium in the sense of Lyapunov is formulated and then successively transformed as additional assumptions are introduced, with special attention focused on isothermal rate-independent plasticity. In the latter case the conditions for stability of a quasi-static process at varying loading are also derived and shown to differ from the respective conditions of stability of equilibrium. The stability conditions are formulated for an arbitrary continuous system with specified boundary conditions as well as for a homogeneous material element embedded in a continuum. Relation between intrinsic instability at the level of a material element and propagation of acceleration waves or strain localization in shear bands is discussed.