Single-support heel-off occurs when the heel of the trailing leg has been lifted from the ground around its toe, while the leading leg is still swinging forward. A similar gait event occurs during human walking, and is crucial to achieve a longer step length and a higher walking speed. In this paper, this crucial gait event is studied, specifically in how it influences the agility and the energy efficiency of bipedal walking. Toward this goal, the concept of limit-cycle bipedal walking which possesses natural and energy-efficient gaits is employed. The aforementioned concept is applied to a flat-foot bipedal model which is developed and actuated by a constant hip torque only during the single-support phase to walk on the ground. The impedance of each ankle is adjusted by using two springs, one at the back-side and the other at the front-side, as well as one damper. In comparison with point/round foot bipedal models, the flat-foot bipedal model produces more versatile limit-cycle gaits comprised of a number of gait series, each of which is a sequence detected among twelve gait postures dictated by the kinetics of the unilateral constraints at the heel, toe, or both. As a result of comprehensive simulations, it is concluded that single-support heel-off significantly improves the agility of bipedal walking because of the increase in the step length and the walking speed. Furthermore, even though limit-cycle gaits including single-support heel-off require higher energy input as compared with gaits excluding such an event, single-support heel-off significantly improves the energy efficiency of bipedal walking since the increase in the step length dominates the increase in the energy input.