The onset of dissipation in thin, tape shaped form of high-Tc superconductors has been studied by the use of differential resistance characteristics in absence or in small magnetic fields. The studies reveal two characteristic currents: the onset current defining a sharp onset of dissipation and the critical current, a parameter of power-law dependencies. A broad range of intermediate currents is characterized by an anomalous saturation at the very low level of dissipation. The observations are interpreted inside the percolation model for the onset of dissipation as a crossover from the range dominated by finiteness of the sample on the scale of current-dependent correlation length to the range of power laws that corresponds to scaling regime of critical phenomena.