Using the SLEX library as the primary tool, we develop a systematic, flexible, and computationally efficient procedure for analyzing multivariate nonstationary time series. The SLEX library is a collection of bases; each of which consists of localized Fourier waveforms. In the problem of signal representation and spectral estimation, one can select, from the set of bases in the SLEX library, the one that best represents the underlying process. Moreover, in discrimination and classification of nonstationary time series, one can select the basis that gives the maximal separation between classes of nonstationary time series. We illustrate the SLEX methods by analyzing multichannel EEGs recorded during an epileptic seizure and during a visual-motor experiment.