Localization-delocalization transitions occur in problems ranging from semiconductor-device physics to propagation of disease in plants and viruses on the internet. Here, we report calculations of localized electronic and vibrational eigenstates for remarkably different, mostly realistic, disordered systems and point out similar characteristics in the cases studied. We show in each case that the eigenstates may be decomposed into exponentially localized islands which may appear in many different eigenstates. In all cases, the decay length of the islands increases only modestly near the localization-delocalization transition; the eigenstates become extended primarily by proliferation (growth in number) of islands near the transition. Recently, microphotoluminescence experiments (Guillet et al 2003 Phys. Rev. B 68 045319) have imaged exciton states in disordered quantum wires, and these bear a strong qualitative resemblance to the island structure of eigenstates that we have studied theoretically.