We study the large-amplitude response of classical molecules to electromagnetic radiation, showing the universality of the transition from linear to nonlinear response and breakup at sufficiently large amplitudes. We demonstrate that a range of models, from the simple harmonic oscillator to the successful Peyrard-Bishop-Dauxois type models of DNA, which include realistic effects of the environment (including damping and dephasing due to thermal fluctuations), lead to characteristic universal behavior: formation of domains of dissociation in driving force amplitude-frequency space, characterized by the presence of local boundary minima. We demonstrate that by simply following the progression of the resonance maxima in this space, while gradually increasing intensity of the radiation, one must necessarily arrive at one of these minima, i.e., a point where the ultrahigh spectral selectivity is retained. We show that this universal property, applicable to other oscillatory systems, is a consequence of the fact that these models belong to the fold catastrophe universality class of Thom's catastrophe theory. This in turn implies that for most biostructures, including DNA, high spectral sensitivity near the onset of the denaturation processes can be expected. Such spectrally selective molecular denaturation could find important applications in biology and medicine.