Diamond Island is a river island in the state of Kentucky, on the Ohio River, on the border between Kentucky and Indiana. It is one of the largest islands in the Ohio River, and possibly the largest unbridged island. It lies about an equal distance from the Kentucky and Indiana shores, but is part of Kentucky. It is located about ten miles southwest of Evansville, Indiana. As with many alluvial islands, there seems to be different estimates on its size: Wikipedia says a half square mile, but a report on the island's sale says it is "estimated to be 1468 acres", which is more than two square miles. My estimates based on Google Maps say that it is about 3 miles at its longest point, and 1 mile at its widest, which, given its diamond shape, does suggest it is close to two square miles.
The island is currently uninhabited, although it does contain several structures, including grain silos, a barn and a ferry dock. Although the edges of the island are forested, most of it is planted with crops, which grow well in the fertile, alluvial soil. It is possible that some of the structures are habitable, but there is apparently no information on this.
There is little information on the history of the island, and much of it seems to be admittedly hearsay. A Works Project Administration book states that the area around the island, and perhaps the island itself, were the home of The Mason Band, a group of river pirates. A more recent book, the "Ohio River Guidebook", states that there is no evidence that European settlers lived on the island, despite records of a school named after the island, which was not necessarily located on the island. In general, while it seems that a large, fertile and easily accessible (in the days of water travel) island like Diamond Island would have been settled, there doesn't seem to be any historical or archaeological evidence that this is so.
While I generally think of the more mystery-laden parts of the United States as being in the far wilderness, in arctic, desert or mountain environments that are outside of normal life, there are still places in the flat, settled, freeway'd part of the United States that are hard to reach, that are invisible to the world, and that are hard to find information about. And Diamond Island is one of these.
Sources about Diamond Island:
http://www.landreport.com/2011/12/sold-kentucky%E2%80%99s-diamond-island/
The WPA Guide to Kentucky: The Bluegrass State By Federal Writers' Project
Ohio River Guidebook By Jerry M. Hay