Multispectral optoacoustic (photoacoustic) tomography (MSOT) is a hybrid modality that can image through several millimeters to centimeters of diffuse tissues, attaining resolutions typical of ultrasound imaging. The method can further identify tissue biomarkers by decomposing the spectral contributions of different photo-absorbing molecules of interest. In this work we investigate the performance of blind source unmixing methods and spectral fitting approaches in decomposing the contributions of fluorescent dyes from the tissue background, based on MSOT measurements in mice. We find blind unmixing as a promising method for accurate MSOT decomposition, suitable also for spectral unmixing in fluorescence imaging. We further demonstrate its capacity with temporal unmixing on real-time MSOT data obtained in-vivo for enhancing the visualization of absorber agent flow in the mouse vascular system.