Making a nuclear bomb is quite the engineering challenge. Everything has to be perfectly tuned for a nuclear bomb to really explode. Not only is a nuclear power plant not designed to be a bomb, it's explicitly designed NOT to be a bomb.
Anti-Nuclear activists generate a lot of hysteria by comparing nuclear power plants to nuclear weapons. Do you realize what would happen if all the fuel burned in a natural gas plant in a year were made into a fuel-air explosive? There would be a big mushroom cloud. But no-one's scared of natural gas plants blowing up, and significantly less effort has been put into designing them to be safe.
Even in the utterly worst case scenario, if every one of the technicians at the least safe nuclear power plant in production conspired to make it fail in the most spectacular possible way - it still wouldn't blow up like a nuclear bomb. At absolute worst, it would melt down.
A melt down is what would theoretically happen in certain obsolete nuclear power plant designs if control of the nuclear chain reaction were completely lost. If that were to happen, the inside of the nuclear power plant would melt, and there would be a radioactive mess to clean up.
Some of these obsolete plants are still in use in the United States. Even though a meltdown is extremely unlikely, and they are explicitly designed to contain any radition in the event of a meltdown, they should still be replaced with more recent designs at some point.
In reasonably modern nuclear power plant designs this simply can't happen. If control stops, or coolant is cut off, or any other drastic failure occurs, a modern nuclear power plant will simply shut itself off. Designs vary in how much effort it takes to restart them after a such a drastic failure, but that's an economic issue, not a safety issue.
In the testing of the Integral Fast Reactor, a test reactor that is representitive of next-generation reactor designs, two drastic coolant failures were simulated in the same day. They unplugged the coolant pumps. They disconnected the turbines so the power couldn't be turned into electricity. In both cases, the reactor just safely stopped, ready to be turned back on when the problem was fixed.
More Info: Wikipedia: Meltdown
The whole concept of Nuclear Waste is widely misunderstood. Current production nuclear reactor designs only use about 1% of the energy content of the nuclear fuel. What comes out isn't waste, but simply fuel that reactors currently in production are not designed to use. That "nuclear waste" can be used in newer plant designs like the Integral Fast Reactor. Once a design like the IFR can't get energy out of nuclear fuel, you have the real nuclear waste - radioative material that cannot be used to make a bomb, and that will stay dangerously radioactive for a reasonably short amount of time.
So... disposing of nuclear waste is easy: you leave it in the storage pool at the power plant until you've built a nuclear plant with a modern fuel cycle to use the fuel in.
Disposing of the final nuclear waste isn't utterly trivial, but it won't require a massive facility under a mountain to keep it out of the ecosystem for hundreds of thousands of years.
With current reactor designs this is unfortunately true. The reactor designs currently in production are unsustainable - because they only use about 1% of the energy content of the fuel. Luckily, more modern reactor designs such as the Integral Fast Reactor don't have this problem; they can use as much as 99.5% of the fuel's energy content.
With 99.5% energy usage, there is enough Uranium for thousands of years. Further, Thorium can be used as a nuclear fuel and there's four times as much Thorium as Uranium in the world.
Thousands of years isn't forever, but it's good enough to stop worrying about the problem for a couple hundred years. For me, that's sustainable enough.
Nuclear Power is an extremely valuable technology. It allows the generation of massive amounts of clean, inexpensive electricity from reasonably common resources. As a result, countries will develop nuclear power technology.
Modern nuclear power technology need not lead towards nuclear weapon technology. Modern designs like the Integral Fast Reactor simply don't lend themselves to the production of pure Uranium 233, Uranium 235 or Plutonium 239, the isotopes that are needed for nuclear weapons.
The key to preventing the proliferation of nuclear weapon technology is the development and sharing of effective and modern proliferation-resistant reactor and process designs. If we force everyone to develop their nuclear power technology from scratch, of course they are going to chose the technological path that gets them nuclear weapons as well - even if it makes their power generation designs less sustainable and efficient.
Consider the nuclear power program in Iran as an example. They are designing their nuclear reactors to run entirely off enriched uranium. This design methodology could produce better power generation results than the current generation nuclear reactors in the United States, but it also has the side effect of making the development of nuclear weapons really easy. If the USA shared our next generation reactor designs like the Integral Fast Reactor with Iran, they would have no excuse to build the massive number of enrichment facilities that they're planning - because the IFR is hundreds of times as efficient for generating power without needing that enrichment and purification capacity.
I guess that makes the myth somewhat true... if we force other countries to develop their own obsolete nuclear technology, that will promote nuclear proliferation. The only way to slow that down is to continue nuclear power research and share the results.