In this paper we consider the problem of classifying non-stationary time series. The method that we introduce is based on the locally stationary wavelet paradigm and seeks to take account of the fact that there may be within-class variation in the signals being analysed. Specifically, we seek to identify the most stable spectral coefficients within each training group and use these to classify a new, previously unseen, time series. In both simulated examples and an aerosol spray example provided by an industrial collaborator, our approach is found to yield superior classification performance when compared against the current state of the art.