The nonlinear stability of laminar sinuously bent streaks is studied for the plane Couette flow at Re=500 in a nearly minimal box and for the Blasius boundary layer at Re_delta=700. The initial perturbations are nonlinearly saturated streamwise streaks of amplitude AU perturbed with sinuous perturbations of amplitude AW. The local boundary of the basin of attraction of the linearly stable laminar flow is computed by bisection and projected in the AU-AW plane providing a well defined critical curve. Different streak transition scenarios are seen to correspond to different regions of the critical curve. The modal instability of the streaks is responsible for transition for AU~25%-27% for the considered flows, where sinuous perturbations of amplitude below AW ~ % 1%-2% are sufficient to counteract the streak viscous dissipation and induce breakdown. The critical amplitude of the sinuous perturbations increases when the streamwise streak amplitude is decreased. With secondary perturbations amplitude A W % 4%, breakdown is induced on stable streamwise streaks with AU = 13%, following the secondary transient growth scenario first examined by Schoppa and Hussain [J. Fluid Mech. 453, 57 (2002)]. A cross-over, where the critical amplitude of the sinuous perturbation becomes larger than the amplitude of streamwise streaks, is observed for streaks of small amplitude AU < 5%-6%. In this case, the transition is induced by an initial transient amplification of streamwise vortices, forced by the decaying sinuous mode. This is followed by the growth of the streaks and final breakdown. The shape of the critical AU-AW curve is very similar for Couette and boundary layer flows and seems to be relatively insensitive to the nature of the edge states on the basin boundary. The shape of this critical curve indicates that the stability of streamwise streaks should always be assessed in terms of both the streak amplitude and the amplitude of spanwise velocity perturbations.