The discovery of quasi-periodic oscillations (QPOs) in magnetar giant flares has opened up prospects for neutron star asteroseismology. However, with only three giant flares ever recorded, and only two with data of sufficient quality to search for QPOs, such analysis is seriously data limited. We set out a procedure for doing QPO searches in the far more numerous, short, less energetic magnetar bursts. The short, transient nature of these bursts requires the implementation of sophisticated statistical techniques to make reliable inferences. Using Bayesian statistics, we model the periodogram as a combination of red noise at low frequencies and white noise at high frequencies, which we show is a conservative approach to the problem. We use empirical models to make inferences about the potential signature of periodic and quasi-periodic oscillations at these frequencies. We compare our method with previously used techniques and find that although it is on the whole more conservative, it is also more reliable in ruling out false positives. We illustrate our Bayesian method by applying it to a sample of 27 bursts from the magnetar SGR J0501+4516 observed by the Fermi Gamma-ray Burst Monitor, and we find no evidence for the presence of QPOs in any of the bursts in the unbinned spectra, but do find a candidate detection in the binned spectra of one burst. However, whether this signal is due to a genuine quasi-periodic process, or can be attributed to unmodeled effects in the noise is at this point a matter of interpretation.
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