The recently developed general materials failure theory is specialized to the two-dimensional state of plane stress. It takes a form that is virtually no more involved than that of the Mises criterion. Yet it remains applicable to the entire range of materials types and thus retains that generality. The Mises form has absolutely no capability for generality. This plane stress form of the new failure theory reveals the existence of three independent modes and mechanisms of failure, not two, not four, purely three. The Mises criterion has one mode of failure. These three modes of failure are fully examined. It is verified that these modes of failure under plane stress conditions are exactly the same as those operative in the three-dimensional case. The simple plane stress form of the failure theory has major appeal and likely use as a teaching tool to introduce failure and to help de-mystify the vitally important general subject of materials failure.