Multispectral optoacoustic tomography (MSOT) utilizes broadband ultrasound detection for imaging biologically-relevant optical absorption features at a range of scales. Due to the multiscale and multispectral features of the technology, MSOT comes with distinct requirements in implementation and data analysis. In this work, we investigate the interplay between scale, which depends on ultrasonic detection frequency, and optical multispectral spectral analysis, two dimensions that are unique to MSOT and represent a previously unexplored challenge. We show that ultrasound frequency-dependent artifacts suppress multispectral features and complicate spectral analysis. In response, we employ a wavelet decomposition to perform spectral unmixing on a per-scale basis (or per ultrasound frequency band) and showcase imaging of fine-scale features otherwise hidden by low frequency components. We explain the proposed algorithm by means of simple simulations and demonstrate improved performance in imaging data of blood vessels in human subjects.