Zero dispersion peaks (ZDPs), which can arise in the fluctuation spectra of noise-driven underdamped oscillators for which the dependence of the eigenfrequency omega on energy E possesses a maximum or minimum, have been investigated by means of analogue electronic experiments. Two different model systems were studied: a tilted Duffing oscillator (TDO); and a bistable superconducting quantum interference device (SQUID) model. It is demonstrated experimentally for the first time that, for strong enough intensity $T$ of Gaussian pseudo-white noise (equivalent to temperature in a thermal system), the shape of the ZDP becomes universal, independent of the system under investigation. Its evolution with T is also shown to exhibit universal features, being governed by a single parameter provided that $T$ exceeds a critical value Tc, below which the ZDP disappears abruptly. The hierarchy of universalities connected to particular types of extrema in omega (E) is discussed. The results are of relevance to underdamped SQUIDs and, in particular, to the recently discovered phenomenon of zero-dispersion stochastic resonance.