Conditions sufficient for instability of deformation are examined for a class of incrementally nonlinear, time independent inelastic solids. Instability of a quasi-static deformation process (path) at varying loading is distinguished from a narrower concept of an unstable equilibrium state. The energy criterion is extended to deformation paths under general assumptions which ensure that the incremental boundary value problem can be given a variational formulation. For a discretized problem, fulfillment of the instability condition along a path is shown to imply either instability of the traversed equilibrium states in a dynamic sense or persisting possibility of quasi-static bifurcation at varying loading. For a continuum, instability at varying loading is interpreted as sensitivity of the incremental deformation to arbitrarily small perturbing forces.