Wavelet transforms are powerful techniques that can decompose time series into both time and frequency components. Their application to experimental data has been hindered by the lack of a straightforward method to handle noise. A noise reduction technique, developed recently for use in wavelet cluster analysis in cosmology and astronomy, is adapted here for time-series data. Noise is filtered using control surrogate data sets generated from randomized aspects of the original time series. The method is a powerful extension of the wavelet transform that is readily applied to the detection of structure in stationary and nonstationary time series.