The Oef category gathers rapidly rotating and evolved O-stars displaying a centrally reversed He II λ 4686 emission line. The origin of the variability of their photospheric and wind spectral lines is debated, with rotational modulation or pulsations as the main contenders. To shed new light on this question, we analysed high-quality and high-cadence Transiting Exoplanet Survey Satellite photometric time series for five Oef stars. We also collected a new time series of spectra for one target (λ Cep) which had been the subject of specific debates in the last years. These observations reveal the variety of Oef behaviours. While space-based photometric data reveal substantial red noise components in all targets, only ζ Pup seems to display a long-lived periodicity. In our sample, stars exhibit a dominant signal at low frequencies but it appears relatively short-lived. This is reminiscent of rotational modulations by transient photospheric spots, though this scenario is challenged by the case of HD 14 442, whose 1.230 d[SUP]-1[/SUP] signal significantly exceeds the critical rotational frequency. In parallel, no evidence of persistent p mode non-radial pulsations is found in either photometry or spectroscopy of the stars, only temporary excitation of g mode pulsations could offer an alternative explanation for the dominant signals. Finally, the revised luminosities of the stars using GAIA-DR2 show that they are not all supergiants as ζ Pup. The question then arises whether the Oef peculiarity denotes a homogeneous class of objects after all.