(former navy nuke here)
When a nuclear power plant is running, the fissions generate a whole boatload of different isotopes. Back in high school, I remember having the distinct impression that a nuclear reactor followed a distinct set of deterministic fissions, producing a neat chain of fission products, mostly due to pictures like this.
It just ain’t so. Everything is smashing around so much that you are likely to find many different isotopes in the mess that is produced, though the products do follow a sort of nonrandom distribution, an interesting curve that some have called the Dolly Parton Curve.
Anyway, among all of that stuff that is generated are many unstable isotopes with vanishingly-small half lives and some with half lives on the order of billions of years.
The heat that runs the power plant comes mostly from the kinetic energy of the fission products being tossed about, with a small percentage (~7%) of full power coming from the decay of the fission products.
See the chart on page 6 of this document (Warning: PDF) for a little insight into the way decay heat works.
Once the reactor is shut down, meaning the neutron population goes below self-sustaining, the reactor still retains that steady-state decay heat of ~7-8%
Immediately, the fast-decaying isotopes start to peter out, resulting in the decay heat dropping, as shown in the curve in the PDF.
After a day, the power is < 1%. After a week, around 0.1%. After a year, around 0.05%. After ten years, .0025%
This means that if you were operating at 1000MW, immediately after shutdown you have to deal with 70MW of energy in the reactor. After a week, you still have 1MW of heat being generated. After a year, half a megawatt. This is why these spent cores still need babysitting.
As far as spent fuel is concerned, you can’t use it anymore in the reactor because it cannot generate enough neutrons to sustain a reaction, regardless of how much decay heat is there. I suppose you could milk it for that remaining megawatt for a year, but you probably want to slap in a new core and get up to full power.
The decay heat is, in fact, being used during full power operations: it is always present and always around 7-8% of your total power output.