We investigate algorithmic conditions which in combination with the Kurdyka-Lojasiewicz inequality assure convergence of non-smooth descent methods to a single critical point. A related question concerns finite length and convergence of discrete sub-gradient trajectories, and convergence of the Talweg. Our findings are somewhat surprising: contrary to the smooth case, convergence and finite length of the trajectories are no longer linked. Additional structural properties of the non-smooth objective function are required to give convergence.