Models of biped walking have demonstrated that stable walking motions are possible without active control. Stability of these walking motions has typically been quantified by studying stability of an associated Poincaré map (orbital stability). However, additional insight may be obtained by examining how perturbations evolve over the short-term (local stability). For example, there may be regions where perturbations actually diverge from the unperturbed trajectory even if over the entire cycle perturbations are dissipated. We present techniques to calculate local stability and demonstrate the utility of these techniques by examining the local stability of the 2D compass biped. These techniques are relevant to the design of controllers to maintain stability in robots, and in understanding how the neuromuscular system maintains stability in humans.