We present a new method, based on fractal analysis, to characterize the output of a physical detector that is in the form of a set of real-valued, discrete physical measurements. We apply the method to gravitational-wave data from the latest observing run of the Laser Interferometer Gravitational-wave Observatory. We show that a measure of the fractal dimension of the main detector output (strain channel) can be used to determine the instrument status, test data stationarity, and identify non-astrophysical excess noise in low latency. When applied to instrument control and environmental data (auxiliary channels) the fractal dimension can be used to identify the origins of noise transients, non-linear couplings in the various detector subsystems, and provide a means to flag stretches of low-quality data.