Cells sense the environment's mechanical stiffness to control their own shape, migration, and fate. To better understand stiffness sensing, we constructed a stochastic model of the "motor-clutch" force transmission system, where molecular clutches link F-actin to the substrate and mechanically resist myosin-driven F-actin retrograde flow. The model predicts two distinct regimes: (i) "frictional slippage," with fast retrograde flow and low traction forces on stiff substrates and (ii) oscillatory "load-and-fail" dynamics, with slower retrograde flow and higher traction forces on soft substrates. We experimentally confirmed these model predictions in embryonic chick forebrain neurons by measuring the nanoscale dynamics of single-growth-cone filopodia. Furthermore, we experimentally observed a model-predicted switch in F-actin dynamics around an elastic modulus of 1 kilopascal. Thus, a motor-clutch system inherently senses and responds to the mechanical stiffness of the local environment.