Context. Transient short-period (<100?s) oscillations have been found in the X-ray light curves of three novae during their super-soft source (SSS) phase and in one persistent SSS. Aims. We pursue an observational approach to determine possible driving mechanisms and relations to fundamental system parameters such as the white dwarf mass. Methods. We performed a systematic search for short-period oscillations in all available XMM-Newton and Chandra X-ray light curves of persistent SSS and novae during their SSS phase. To study time evolution, we divided each light curve into short time-segments and computed power spectra. We then constructed a dynamic power spectrum from which we identified transient periodic signals even when only present for a short time. We base our confidence levels on simulations of false-alarm probability for the chosen oversampling rate of 16, corrected for multiple testing based on the number of time segments. From all time segments of each system, we computed fractions of time when periodic signals were detected. Results. In addition to the previously known systems with short-period oscillations, RS?Oph (35?s), KT?Eri (35?s), V339?Del (54?s), and Cal?83 (67?s), we found one additional system, LMC?2009a (33?s), and also confirm the 35?s period from Chandra data of KT?Eri. The oscillation amplitudes are of about <15% of the respective count rates and vary without any clear dependence on the X-ray count rate. The fractions of the time when the respective periods were detected at 2σ significance (duty cycle) are 11.3%, 38.8%, 16.9%, 49.2%, and 18.7% for LMC?2009a, RS?Oph, KT?Eri, V339?Del, and Cal?83, respectively. The respective highest duty cycles found in a single observation are 38.1%, 74.5%, 61.4%, 67.8%, and 61.8%. Conclusions. Since fast rotation periods of the white dwarfs as origin of these transient oscillations are speculative, we concentrate on pulsation mechanisms. We present initial considerations predicting the oscillation period to scale linearly with the white dwarf radius (and thus mass), weakly with the pressure at the base, and luminosity. Estimates of the size of the white dwarf could be useful for determining whether these systems are more massive than typical white dwarfs, and thus whether they are growing from accretion over time. Signs of such mass growth may have implications for whether some of these systems are attractive as Type Ia supernova progenitors.
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