I'll back up a bit...in naval reactors no plutonium is used or produced. I can't be sure of what civilian reactors have, though I'm 98% positive that plutonium cannot be produced by most US reactors, for the reasons barfo stated. In military reactors, the spent cores are wholly Uranium Oxide, which is what it came out of the ground as. Sure, it has been "enriched" (purifying down to a higher percentage of one naturally occurring isotope over another--I can't give the percentages) but it's not like when Uranium oxide is used up, it becomes Plutonium or something "really bad". The cores are Uranium oxide coming from the ground, and uranium oxide going back in.
Chernobyl was a fundamentally different type of reactor than ours. We used pressurized water reactors in the west, while the Soviets used liquid graphite. In our reactors, if something bad happens and the core starts heating up, the power actually goes down, preventing a meltdown. In the Soviet ones, power goes up, so heat produced goes up, so core temperature goes up, which causes power to go up, etc. in an increasing death spiral. All reactors have "scram" controls put in to shut down the reactor if a certain power level or temperature is reached. In Chernobyl's case, those controls were disabled b/c the operators were preparing to perform a test.
The parts to worry about in disassembly of spent cores is the coolant and Ion Exchanger resin. But as with just about everything, dangerous things are highly mitigated through proper procedural compliance. In over 50 years of having every submarine in the Navy being powered by nuclear reactors there hasn't been one accident, meltdown, etc.
