Elastic–rod models of DNA have offered an alternative method for studying the macroscopic properties of the molecule. An essential component of the modelling effort is to identify the biologically accessible, or stable, solutions. The underlying variational structure of the elastic–rod model can be exploited to derive methods that identify stable equilibrium configurations. We present two methods for determining the stability of the equilibria of elastic–rod models: the conjugate–point method and the distinguished–diagram method. Additionally, we apply these methods to two intrinsically curved DNA molecules: a DNA filament with an A–tract bend and a DNA minicircle with a catabolite gene activator protein binding site. The stable solutions of these models provide visual insight into the three–dimensional structure of the DNA molecules.