We will present our experience on applying discrete group theory arguments to study the nonlinear Schrödinger equation. First, we will present the angular pseudomomentum theory, i.e., a set of results that allows to classify and predict some properties of the stationary solutions of such equation with discrete rotational symmetry. Next, we will use these arguments to study the dynamical evolution of symmetrical solutions. Finally, we will use all this knowledge in the study of the dynamical behaviour of phase singularities of an optical field, a topic that is commonly enclosed in a separated branch of optics called "nonlinear singular optics".