The split main sequences (MSs) and extended MS turnoffs (eMSTOs) detected in a few young clusters have demonstrated that these stellar systems host multiple populations differing in a number of properties such as rotation and, possibly, age. We analyse Hubble Space Telescope photometry for 13 clusters with ages between ˜40 and ˜1000 Myr and of different masses. Our goal is to investigate for the first time the occurrence of multiple populations in a large sample of young clusters. We find that all the clusters exhibit the eMSTO phenomenon and that MS stars more massive than ˜1.6 M_{⊙} define a blue and a red MS, with the latter hosting the majority of MS stars. The comparison between the observations and isochrones suggests that the blue MSs are made of slow-rotating stars, while the red MSs host stars with rotational velocities close to the breakup value. About half of the bright MS stars in the youngest clusters are H α emitters. These Be stars populate the red MS and the reddest part of the eMSTO, thus supporting the idea that the red MS is made of fast rotators. We conclude that the split MS and the eMSTO are a common feature of young clusters in both Magellanic Clouds. The phenomena of a split MS and an eMSTO occur for stars that are more massive than a specific threshold, which is independent of the host-cluster mass. As a by-product, we report the serendipitous discovery of a young Small Magellanic Cloud cluster, GALFOR 1.