Exactly as the title says. What is the archaeological evidence that Sodom and Gomorrah existed and were destroyed by fire from the heavens (Genesis 19:24)?
The search for the cities of Sodom and Gomorrah have been a common archaeological pursuit over the years. There have been some proposals, and I do recall seeing a documentary about a couple such sites, near the Dead Sea, which may be very likely the Biblical cities stated. It is difficult to say for absolute certain whether these are the cities mentioned in the Bible or not, but there are some interesting arguments for their identification as such.
The documentary that I saw was on the History Channel, and mentioned the sites named Bab edh-Dhra and Numeira as candidates. They existed near each other and both were destroyed, somehow, around the same time.
There are other cities that have been proposed, however. You can see a quick writeup on Wikipedia here.
It's worth noting that it would appear from the text that the Cities of the Plain (which include Sodom and Gomorrah) were destroyed by some type of meteor impact. If so, it's not likely that there was much left behind for modern archaeologists to find, especially considering the primitive state of construction technology back then.
If a meteor were to wipe out a modern metropolis, full of reinforced concrete and steel, I'd expect to be able to come back a few thousand years later and find evidence that there was once a city there. But a bunch of communities with buildings made of wood and clay wouldn't stand a chance of surviving an impact event with a kinetic yield measured in increments usually reserved only for describing nuclear weapons, even as ruins.