From my observation, people seem to think that "nuclear waste" is green glowing goo that turns people into mutants, and it's all the same no matter where it comes from. Thank Hollywood. Waste from nuclear power plants is basically expended fuel rods. They are radioactive, but the radioactivity is contained to an extent (the uranium oxide that is used as fuel is encased in a zirconium alloy). It's not something you'd want to hold in your hand obviously, but it's not *that* dangerous. These are typically stored in dry casks, that are filled with helium or some other inert gas, to keep chemical reactions from breaking down the zirconium alloy around the fuel pellets.
The REALLY nasty nuclear waste (that is typically partially or mostly liquids and is stored in underground tanks at places like Hanford), does not come from nuclear power plants at all. It came from making plutonium for bombs. This stuff is nasty...not only is it extremely radioactive, but it's also *chemically* active (usually highly acidic due to nitric acid being used during the plutonium making process), and also mixed with all kinds of nasty toxic organic materials (another component of the plutonium making process is tributyl phosphate dissolved in kerosene). Take that, mix with nitric acid, mix with all kinds of radioactive salts, and you get something that is very nasty. The process is detailed here: https://en.wikipedia.org/wiki/...
Although keep in mind, PUREX is the latest process, before it was developed there were earlier messier processes that created more waste, and in places like Hanford all this stuff gets mixed together, and who knows what chemical reactions take place in there.
But most people don't really know any of this, they only know of "nuclear waste" that Hollywood has told them about.