Although anecdotal evidence from my apartment strongly suggests otherwise, apparently the cockroach is not the hardiest of all insects, nor is it the critter most likely to survive a nuclear holocaust. Here's Slate's Daniel Riley:
In 1962, H. Bentley Glass, a Johns Hopkins geneticist, told the New York Times that in the event of nuclear war, "the cockroach, a venerable and hardy species, will take over the habitations of the foolish humans, and compete only with other insects or bacteria." But studies over the last half-decade, such as those conducted by the United Nations Scientific Committee on the Effects of Atomic Radiation, have found that these "other insects" are more likely to reign in the age after humans; the cockroach might, in fact, be one of the first bugs to go.
More recently, the television show MythBusters tested the effects of radiation on several kinds of insects and discovered that tiny flour beetles were the hardiest—with some surviving a dose of 100,000 rads. ... Organisms that aren't classified as animals are even better-equipped to handle a nuclear fallout: certain bacteria, protozoa, mosses, and algae might thrive long after roaches and flour beetles bite the dust.
That said, the cockroach has been skittering around since 300 million B.C.—long before dinosaurs—and can survive for more than a month without a head, so I'll still give it better-than-even odds.