We conducted a 12-month monitoring campaign of 33 T Tauri stars (TTS) in Taurus. Our goal was to monitor objects that possess a disc but have a weak H{alpha} line, a common accretion tracer for young stars, in order to determine whether they host a passive circumstellar disc. We used medium-resolution optical spectroscopy to assess the accretion status of the objects and to measure the H{alpha} line. We found no convincing examples of passive discs: only transition disc and debris disc systems in our sample are non-accreting. Among accretors, we found no example of flickering accretion, leading to an upper limit of 2.2 per cent on the duty cycle of accretion gaps, assuming that all accreting TTS experience such events. When combining literature results with our observations, we found that the reliability of traditional H{alpha}-based criteria to test for accretion is high but imperfect, particularly for low-mass TTS. We found a significant correlation between stellar mass and the full width at 10 per cent of the peak (W_10_) of the H{alpha} line that does not seem to be related to variations in free-fall velocity. Finally, our data revealed a positive correlation between the H{alpha} equivalent width and its W_10_, indicative of a systematic modulation in the line profile whereby the high-velocity wings of the line are proportionally more enhanced than its core when the line luminosity increases. We argue that this supports the hypothesis that the mass accretion rate on the central star is correlated with the H{alpha} W_10_ through a common physical mechanism.