We have performed three searches for high-frequency signals in the solar neutrino flux measured by the Sudbury Neutrino Observatory (SNO), motivated by the possibility that solar $g$-mode oscillations could affect the production or propagation of solar $^8$B neutrinos. The first search looked for any significant peak in the frequency range 1/day to 144/day, with a sensitivity to sinusoidal signals with amplitudes of 12% or greater. The second search focused on regions in which $g$-mode signals have been claimed by experiments aboard the SoHO satellite, and was sensitive to signals with amplitudes of 10% or greater. The third search looked for extra power across the entire frequency band. No statistically significant signal was detected in any of the three searches. ; Comment: 37 pages, 17 figures, submitted to ApJ