Push recovery is a vital aspect of balance stability control in biped robots. In this work, the response of a biped system to unexpected external perturbations is analyzed for different tasks and controllers using stability criteria based on balanced and steppable regions. The steppable region for a given step length and the balanced regions for single and double support contacts are constructed for a biped robot using optimization with its system dynamics, kinematic limits, actuation limits, and contact interactions with the environment. The regions are compared with those of a human subject to demonstrate that human gait exhibits unbalanced (but steppable) phases largely absent in robotic gait. These regions are also applied to a comparative analysis against capturability, where the computed steppable region is significantly larger than the capture region of an equivalent reduced-order model. The stability regions are also used to compare the performance of controllers during a double support balancing task. The implemented hip, knee, and ankle strategy-based controller led to improved stabilization — i.e., decreased foot tipping and time required to balance — relative to an existing hip and ankle controller and a gyro feedback controller. The proposed approaches are applicable to the analysis of any bipedal task and stability controller in general.