Generalized joint signal representations (JSRs) extend the scope of joint time-frequency representations (TFRs) to a richer class of nonstationary signals, but their use, just as in the case of TFRs, has been primarily limited to qualitative, exploratory data analysis. To exploit their potential more fully, JSR-based statistical signal processing techniques need to be developed that can be successfully applied in real-world problems. In this paper, we present an optimal detection framework based on arbitrary generalized quadratic JSRs, thereby making it applicable in a wide variety of detection scenarios involving nonstationary stochastic signals, noise and interference. For any given class of generalized JSRs, we characterize the corresponding class of detection scenarios for which such JSRs constitute canonical detectors, and derive the corresponding JSR-based detectors. Our formulation also yields a very useful subspace-based interpretation in terms of corresponding linear JSRs that we exploit to design optimal detectors based on only partial signal information.