Connecticut, like a number of other states, has its own panhandle. However, due to circumstances surrounding its creation, it created a mysterious region in New York State, known as the Oblong. Here is the story of the creation of both, in brief:
Under a new agreement of November 28, 1683 the boundary between Connecticut and New York was generally recognized as a line parallel to and twenty miles from the Hudson River north to the Massachusetts line. However, New York, acknowledging most of Connecticut’s settlements in Fairfield County, gave up a claims to a 61,660 acre rectangle east of the Byram River, which became the area sometimes referred to as Connecticut’s “panhandle” or the “handle of the cleaver”. In return, Connecticut gave up its claims to Rye and ceded to New York a strip of land 580 rods (1.81 miles) wide “equivalent” to the area of the panhandle that extended north from Ridgefield along Dutchess, Putnam, and Westchester Counties, New York, to the Massachusetts line.
While this strange region of New York is no longer particularly relevant (aside from the marker pictured below, indicating its existence), it is primarily known about within genealogical circles, as families that hail from the Oblong find it to be a genealogical black hole. Apparently, families from this region are ignored by both states, leading to some family tree dead ends.