First of all, Chernobyl simply cannot happen in an American reactor. American reactors are designed quite differently from Russian reactors. In the Russian reactors, components are almost built to fail. For example, at Chernobyl, one wall was designed to fall down easily so that it could be raised back up easily! Another example is the fail-safe design for the damping rods. Damping rods are what control the rate of reaction. If you draw the damping rods out of the pile, the reaction speeds up. If you put them back in, the reaction slows down. If you put the rods all the way in, then you have "scrammed" the reactor, and it will shut down completely (this is usually undesirable during normal operation, because the build-up of a certain noble gas (argon, maybe?) makes it difficult to restart the reactor: you have to wait 72 hours or something like that). In Russian reactors, the damping rods often come into the pile from below. So gravity will tend to push the rods down and out of the pile, creating a dangerous runaway reaction. In most American reactors, the rods come down in from the top as a fail-safe mechanism. If something happens to the control mechanism, the rods will drop down into the pile and scram the reactor.
Three-Mile Island was extremely mild as reactor accidents go. No one died, and few if any were injured.
The Navy went down and investigated the Thresher's reactor hull when it went down, and discovered it intact. Therefore, whatever caused it to go down, it was not a nuclear accident.
I claim nuclear power is safer than coal power by a long shot. The main argument for this is simple: mining. Coal mining is the most dangerous occupation in the world. That is because it is a deep-shaft sort of mining: you have to dig deep for coal. Uranium is mined in an open pit, which is considerably safer.
There are two common misconceptions about nuclear power. One is that a nuclear reactor can blow up like an atomic bomb. This is nonsense. No nuclear reactor in existence can blow up like a bomb. The reason? You have to use a much higher grade of uranium (or plutonium) for a bomb. In other words, it has to be much purer in order to explode like a bomb. Reactors don't need, and don't use such high-grade uranium. They use "lower octane", so to speak. So no fears on that score.
Another misconception about nuclear reactors is that living near one will cause you to have an abnormally high exposure to radiation. This is simply not true. It is a fact that, all other things being equal, a man will get more radiation exposure on one plane trip than he would get by working in a nuclear power plant for a year. Surprised? It is actually pilots that get more radiation than people who work in nuclear power plants; they get the solar radiation. In fact, your local nuclear power plant worker does not get measurably more radiation than the average Joe Blow on the street.
Finally, I would dispel one more myth. Microwave ovens have been accused of giving people radiation. This is simply ludicrous. While I'd be wary of the really old ovens that work with the door open, such an opinion just shows an ignorance of physics. Microwave ovens work by exciting water molecules, which in turn heat up the surrounding molecules. There is no radioactive decay going on at all. The most danger you are in, even if you have an old oven, is getting a little "excited" yourself. Interesting little story: microwaves, thought of as a mechanism for cooking things, were discovered by a man who had a raw egg in his pocket (don't ask me why), and was working, I think, near some transformer or radio station. Whichever station it was gave off some microwaves, and the man found that the egg in his pocket was cooking. The man suffered no harm by this discovery.