Abstract <p>The paper considers the universal signs of the phenomenon of dynamic failure of metals, their dispersion, and fully developed turbulence. It is shown that the critical behavior of nonlinear systems of various physical nature is associated with the phenomena of self-organization, dynamic chaos, and stochastic instability. This behavior is preceded by the appearance of scale-invariant spatiotemporal dissipative hierarchical structures in systems which arise through a cascade of bifurcations. The results of the study of a number of nonequilibrium processes have shown that the kinetics of the occurrence of dissipative structures has the properties of a succession of events, the time intervals between which have statistically self-similar distributions. The singularity spectrum of a multifractal measure describing a series of expectation times is a characteristic of the considered nonlinear processes. From a physical standpoint, the singularity spectrum <i>f</i>(α) can be considered as a quantitative characteristic of nonequilibrium systems undergoing bifurcation transitions.</p>