We report on the thinning mechanisms of supported films of surfactant (nTAB) solutions above the critical micellar concentration. The films are formed by pressing an oil drop immersed in an aqueous surfactant solution on a silica surface. Depending on the length of the carbon chain of the surfactant and its concentration, two modes of destabilization of the stratified films are observed. The first one proceeds by heterogeneous nucleation, characterized by the lateral expansion of the domain of lower thickness as evidenced long ago in suspended micellar films. In addition, the simultaneous stepwise thinning of several domains, called spinodal stratification, is observed here in supported films. We measure the time evolution of the thickness of the films, and we discuss the selection mechanism of each destabilization mode.