High-resolution animal location data are increasingly available, requiring analytical approaches and statistical tools that can accommodate the temporal structure and transient dynamics (non-stationarity) inherent in natural systems. Traditional analyses often assume uncorrelated or weakly correlated temporal structure in the velocity (net displacement) time series constructed using sequential location data. We propose that frequency and time-frequency domain methods, embodied by Fourier and wavelet transforms, can serve as useful probes in early investigations of animal movement data, stimulating new ecological insight and questions. We introduce a novel movement model with time-varying parameters to study these methods in an animal movement context. Simulation studies show that the spectral signature given by these methods provides a useful approach for statistically detecting and characterizing temporal dependency in animal movement data. In addition, our simulations provide a connection between the spectral signatures observed in empirical data with null hypotheses about expected animal activity. Our analyses also show that there is not a specific one-to-one relationship between the spectral signatures and behavior type and that departures from the anticipated signatures are also informative. Box plots of net displacement arranged by time of day and conditioned on common spectral properties can help interpret the spectral signatures of empirical data. The first case study is based on the movement trajectory of a lion (Panthera leo) that shows several characteristic daily activity sequences, including an active-rest cycle that is correlated with moonlight brightness. A second example based on six pairs of African buffalo (Syncerus caffer) illustrates the use of wavelet coherency to show that their movements synchronize when they are within approximately 1 km of each other, even when individual movement was best described as an uncorrelated random walk, providing an important spatial baseline of movement synchrony and suggesting that local behavioral cues play a strong role in driving movement patterns. We conclude with a discussion about the role these methods may have in guiding appropriately flexible probabilistic models connecting movement with biotic and abiotic covariates.