Continuation methods are used to examine the static and dynamic postbuckled behavior of a uniaxially loaded, simply supported plate. Continuation methods have been extensively used to study problems in mathematics and physics; however, they have not been as widely applied to problems in engineering. When paired with a Galerkin approximation, continuation methods are shown to be well suited to solving nonlinear buckling problems. In addition to providing a robust solution method for nonlinear equations, the linearized Jacobians from the continuation steps will contain natural frequency and mode shape information for mechanical systems (provided inertia terms are included). Results for the primary buckling branch are compared to previously published results. Using the open-source continuation package Auto, stable, remote secondary buckling branches were discovered. These secondary stable equilibrium persist even in the presence of geometric imperfections and their existence is confirmed by experiment.