Frequency instabilities are a major source of errors in quantum devices. This Letter investigates frequency fluctuations in a surface acoustic wave (SAW) resonator, where reflection coefficients of 14 SAW modes are measured simultaneously for more than 7 h. We report two distinct noise characteristics. Multimode frequency noise caused by interactions with two-level system (TLS) defects shows significant degrees of correlation that diminish with increased detuning. This finding agrees with the current understanding of the parasitic TLS behavior as one of the dominant noise sources in quantum devices. In addition to the TLS-induced noise, we observe strong anomalous frequency fluctuations with slow, anticorrelated dynamics. These noise bursts resemble signatures of cosmic radiation observed in superconducting quantum systems.