This paper studies the displacement and efficiency of a Purcell’s three-link microswimmer in low Reynolds number regime, capable of moving by the implementation of a motion primitive or gait. An optimization is accomplished attending to the geometry of the swimmer and the motion primitives, considering the shape of the gait and its amplitude. The objective is to find the geometry of the swimmer, amplitude and shape of the gaits which make optimal the displacement and efficiency, in both an individual way and combined (the last case will be referred to as multiobjective optimization). Three traditional gaits are compared with two primitives proposed by the authors and other three gaits recently defined in the literature. Results demonstrate that the highest displacement is obtained by the Tam and Hosoi optimal velocity gait, which also achieves the best efficiency in terms of energy consumption. The rectilinear and Tam and Hosoi optimal efficiency gaits are the second optimum primitives. Regarding the multiobjective optimization and considering the two criteria with the same weight, the optimum gaits turn out to be the rectilinear and Tam and Hosoi optimal efficiency gaits. Thus, the conclusions of this study can help designers to select, on the one hand, the best swimmer geometry for a desired motion primitive and, on the other, the optimal method of motion for trajectory tracking for such a kind of Purcell’s swimmers depending on the desired control objective.