Sarcomeres are the basic force generating units of striated muscles and consist of an interdigitating arrangement of actin and myosin filaments. While muscle contraction is usually triggered by neural signals, which eventually set myosin motors into motion, isolated sarcomeres can oscillate spontaneously between a contracted and a relaxed state. We analyze a model for sarcomere dynamics, which is based on a force-dependent detachment rate of myosin from actin. Our numerical bifurcation analysis of the spontaneous sarcomere dynamics reveals notably Hopf bifurcations, canard explosions, and gluing bifurcations. We discuss possible implications for experiments.