The effect of underwater acoustic propagation on the cyclostationary features of communication signals is modeled and analyzed. Two kinds of channels are considered: the multiscale-multilag channel, over which mobile and wideband acoustic systems usually communicate, and the dispersive channel resulting from low-frequency modal propagation in shallow water. It is shown that multiscale-multilag channels transform cyclostationary signals into a sum of velocity and acceleration-dependent time-warped cyclostationary processes. This time-warping is carefully taken into account to efficiently recover the cyclostationary features. On the other hand, it is found that low-frequency dispersive channels preserve the original periodicity but attenuate the shorter cycles and spread the correlations. To illustrate the theoretical results, applications with simulated and real data are also presented. Specifically, the problem of estimating time-varying Doppler scales is addressed for multiscale-multilag channels as well as the detection of signals with unique cyclostationary signatures. The example of blind symbol-rate estimation applied to covert communications in dispersive channels is also discussed. Special attention is paid to PSK, QAM, OFDM and DSSS signals. Accompanying supplementary material provides the MATLAB code used for the estimation and detection examples.