Last decade saw the creation of a number of directional representation dictionaries that desire to address the weaknesses of the classical wavelet transform that arise due to its limited capacity for the analysis of edge-like features of two-dimensional signals. Salient features of these dictionaries are directional selectivity and anisotropic treatment of the axes, achieved through the parabolic scaling law. In this paper we will examine the adequacy of such dictionaries for the analysis of edge- and corner-like features of 2D regions through a comprehensive framework for directional parabolic dictionaries, called the continuous parabolic molecules. This work builds on a family of earlier studies and aims to give a broader perspective through the level of generality.