Context. Glitches are rare spin-up events that punctuate the smooth slow-down of the rotation of pulsars. For the Vela pulsar and PSR J0537−6910, their large glitch sizes and the times between consecutive events have clear preferred scales (Gaussian distributions), contrary to the handful of other pulsars with enough glitches for such a study. Moreover, PSR J0537−6910 is the only pulsar that shows a strong positive correlation between the size of each glitch and the waiting time until the following one. Aims. We attempt to understand this behaviour through a detailed study of the distributions and correlations of glitch properties for the eight pulsars with at least ten detected glitches. Methods. We modelled the distributions of glitch sizes and of the times between consecutive glitches for the eight pulsars with at least ten detected events. We also looked for possible correlations between these parameters and used Monte Carlo simulations to explore two hypotheses that could explain why the correlation so clearly seen in PSR J0537−6910 is absent in other pulsars. Results. We confirm the above results for Vela and PSR J0537−6910, and verify that the latter is the only pulsar with a strong correlation between glitch size and waiting time to the following glitch. For the remaining six pulsars, the waiting time distributions are best fitted by exponentials, and the size distributions are best fitted by either power laws, exponentials, or log-normal functions. Some pulsars in the sample yield significant Pearson and Spearman coefficients (r_p and r_s) for the aforementioned correlation, confirming previous results. Moreover, for all except the Crab pulsar, both coefficients are positive. For each coefficient taken separately, the probability of this happening is 1/16. Our simulations show that the weaker correlations in pulsars other than PSR J0537−6910 cannot be due to missing glitches that are too small to be detected. We also tested the hypothesis that each pulsar may have two kinds of glitches, namely large, correlated ones and small, uncorrelated ones. The best results are obtained for the Vela pulsar, which exhibits a correlation with r_p = 0.68 (p-value = 0.003) if its two smallest glitches are removed. The other pulsars are harder to accommodate under this hypothesis, but their glitches are not consistent with a pure uncorrelated population either. We also find that all pulsars in our sample, except the Crab pulsar, are consistent with the previously found constant ratio between glitch activity and spin-down rate, ν̇_g/|ν̇| = 0.010±0.001, even though some of them have not shown any large glitches. Conclusions. To explain these results, we speculate except in the case of the Crab pulsar, that all glitches draw their angular momentum from a common reservoir (presumably a neutron superfluid component containing ≈1% of the star’s moment of inertia). However, two different trigger mechanisms could be active, a more deterministic one for larger glitches and a more random one for smaller ones.