The integration of soft actuating materials within origami-based mechanisms is a novel method to amplify the actuated motion and tune the compliance of systems for low stiffness applications. Origami structures provide natural flexibility given the extreme geometric difference between thickness and length, and the energetically preferred bending deformation mode can naturally be used as a form of actuation. However, origami fold patterns that are designed for specific actuation motions and mechanical loading scenarios are needed to expand the library of fold-based actuation strategies. In this study, a recently developed optimization framework for maximizing the performance of compliant origami mechanisms is utilized to discover optimal actuating fold patterns. Variant patterns are discovered through exploring different symmetries in the input and output conditions of the optimization problem. Patterns designed for twist (rotational symmetry) yield significantly better performance, in terms of both geometric advantage and energy requirements, than patterns exhibiting vertical reflection symmetries. The mechanical energy requirements for each design are analyzed and compared for both the small and large applied displacement regimes. Utilizing the patterns discovered through optimization, the multistability of the actuating arms is demonstrated empirically with a paper prototype, where the stable configurations are accessed through local vertex pop-through instabilities. Lastly, the coupled mechanics of fold networks in these actuators yield useful macroscopic motions and can achieve stable shape change through accessing the local vertex instabilities. This survey of origami mechanisms, energy comparison, and multistability characterization provides a new set of designs for future integration with soft actuating materials.