Detection and analysis of astrophysical sources from the forthcoming MUSE instrument is of greatest challenge mainly due to the high noise level and the three-dimensional translation variant blur effect of MUSE data. In this work, we use some realistic hypotheses of MUSE to reformulate the data convolution model into a set of linear mixing models corresponding to different, disjoint spectral frames. Based on the linear mixing models, we propose a spatial-spectral unmixing (SSU) algorithm to detect and characterize the galaxy spectra. In each spectral frame, the SSU algorithm identifies the pure galaxy regions with a theoretical guarantee, and estimate spectra based on a sparse approximation assumption. The full galaxy spectra can finally be recovered by concatenating the spectra estimates associated with all the spectral frames. The simulations were performed to demonstrate the efficacy of the proposed SSU algorithm.