There are several named islands in Middle-earth. The most well-known one was (of course) Númenor, which was sunk in the Change of the World near the end of the Second Age.
Tol Eressëa was used by the Vala Ulmo to ferry Elves from Middle-earth to Valinor. Many of the Eldar continued to live on the island.
There were three islands of the coast of Forlindon that were surviving remnants of sunken Belariand: Himling (originally the Hill of Himring); Tol Fuin; and Tol Morwen (where was located the grave of Túrin and Morwen. As far as I know, none of these islands were inhabited.
Tolfalas was a large island in the Bay of Belfalas. Probably inhabited by folk of Gondor, but Tolkien didn't write much about it.
There seems to be an island near the eastern end of the Inland Sea of Rhûn. If so, it might have been inhabited by anyone from fisherfolk to pirates.
South and east of the lands of Harad was the continent alternately referred to as South Land or Dark Land. The Númenóreans allegedly never colonized it, though it seems likely that other people of Middle-earth would have sailed there at some later time and started settlements. It might have broken up at the Change of the World into smaller lands/islands (modern Australia, Antarctica and Indonesia).
Further east was the continent of the Land of the Sun, also said to be uninhabited. Also presumably destroyed in the Change of the World.