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The estimation of sparse shallow-water acoustic communication channels and the impact of estimation performance on the equalization of phase coherent communication signals are investigated. Given sufficiently wide transmission bandwidth, the impulse response of the shallow-water acoustic channel is often sparse as the multipath arrivals become resolvable. In the presence of significant surface waves, the multipath arrivals associated with surface scattering fluctuate rapidly over time, in the sense that the complex gain, the arrival time, and the Dopplers of each arrival all change dynamically. A sparse channel estimation technique is developed based on the delay-Doppler-spread function representation of the channel. The delay-Doppler-spread function may be considered as a first-order approximation to the rapidly time-varying channel in which each channel component is associated with Doppler shifts that are assumed constant over an averaging interval. The sparse structure of the delay-Doppler-spread function is then exploited by sequentially choosing the dominant components that minimize a least squares error. The advantage of this approach is that it captures both the channel structure as well as its dynamics without the need of explicit dynamic channel modeling. As the symbols are populated with the sample Dopplers, the increase in complexity depends on the channel Doppler spread and can be significant for a severely Doppler-spread channel. Comparison is made between nonsparse recursive least squares (RLS) channel estimation, sparse channel impulse response estimation, and estimation using the proposed approach. The results are demonstrated using experimental data. In training mode, the proposed approach shows a 3-dB reduction in signal prediction error. In decision-directed mode, it improves significantly the robustness of the performance of the channel-estimate-based equalizer against rapid channel fluctuations.