Bilinear time-frequency representations (TFRs) and time-scale representations (TSRs) are potentially very useful for detecting a nonstationary signal in the presence of nonstationary noise or interference. As quadratic signal representations, they are promising for situations in which the optimal detector is a quadratic function of the observations. All existing time-frequency formulations of quadratic detection either implement classical optimal detectors equivalently in the time-frequency domain, without fully exploiting the structure of the TFR, or attempt to exploit the nonstationary structure of the signal in an ad hoc manner. We identify several important nonstationary composite hypothesis testing scenarios for which TFR/TSR-based detectors provide a "natural" framework; that is, in which TFR/TSR-based detectors are both optimal and exploit the many degrees of freedom available in the TFR/TSR. We also derive explicit expressions for the corresponding optimal TFR/TSR kernels. As practical examples, we show that the proposed TFR/TSR detectors are directly applicable to many important radar/sonar detection problems. Finally, we also derive optimal TFR/TSR-based detectors which exploit only partial information available about the nonstationary structure of the signal.