Several template models have been developed to facilitate the analysis of limit-cycles for quadrupedal locomotion. The parameters in the model are usually fixed; however, biology shows that animals change their leg stiffness according to the locomotion velocity, and this adaptability invariably affects the stability of the gait. This paper provides an analysis of the influence of this variable leg stiffness on the stability of different quadrupedal gaits. The analysis exploits a simplified quadrupedal model with compliant legs and shoulder joints represented as torsional springs. This model can reproduce the most common quadrupedal gaits observed in nature. The stability of such emerging gaits is then checked. Afterward, an optimization process is used to search for the system parameters that guarantee maximum gait stability. Our study shows that using the highest feasible leg swing frequency and adopting a leg stiffness that increases with the speed of locomotion noticeably improves the gait stability over a wide range of horizontal velocities while reducing the oscillations of the trunk. This insight can be applied in the design of novel elastic quadrupedal robots, where variable stiffness actuators could be employed to improve the overall locomotion behavior.