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The detection of climate change and its attribution to the corresponding underlying processes is challenging because signals such as trends and shifts are superposed on variability arising from the memory within the climate system. Statistical methods used to characterize change in time series must be flexible enough to distinguish these components. Here we propose an approach tailored to distinguish these different modes of change by fitting a series of models and selecting the most suitable one according to an information criterion. The models involve combinations of a constant mean or a trend superposed to a background of white noise with or without autocorrelation to characterize the memory, and are able to detect multiple changepoints in each model configuration. Through a simulation study on synthetic time series, the approach is shown to be effective in distinguishing abrupt changes from trends and memory by identifying the true number and timing of abrupt changes when they are present. Furthermore, the proposed method is better performing than two commonly used approaches for the detection of abrupt changes in climate time series. Using this approach, the so-called hiatus in recent global mean surface warming fails to be detected as a shift in the rate of temperature rise but is instead consistent with steady increase since the 1960s/1970s. Our method also supports the hypothesis that the Pacific decadal oscillation behaves as a short-memory process rather than forced mean shifts as previously suggested. These examples demonstrate the usefulness of the proposed approach for change detection and for avoiding the most pervasive types of mistake in the detection of climate change.