Islands are divided according to their phase space structure—resonant islands and tangle islands are considered. It is proved that in the near-integrable limit these correspond to two distinct sets, hence that in general their definitions are not trivially equivalent. It is demonstrated and proved that accelerator modes of the standard map and of the web map are necessarily of the tangle island category. These islands have an important role in determining transport—indeed it has been demonstrated in various works that stickiness to these accelerator modes may cause anomalous transport even for initial conditions starting in the ergodic component.