My understanding is that in commercial nuclear reactor operations, fuel rods are not used up to the point where they're fully depleted and unable to support fission, but are replaced while they still contain an appreciable amount of fissionable isotopes, to ensure that the reactor stays in a stable operating regime at all times.
However, would it be possible, for any reason, to run a nuclear reactor without replenishing the fuel as long as any useful energy at all could be produced, that is, actually "running the fuel rods dry" (or, more formally, until criticality is irrecoverably lost and decay heat is the only remaining output left, at which point the reactor is essentially just a fancy spent fuel pool)?
Could such an end state be safely achieved, with power just gradually fading away while the control rods extend out more and more to compensate for the declining reactivity? Or would unstable and potentially dangerous operation ensue, as in Chernobyl where operator response to a poisoned core started a catastrophic chain of events?
The question is motivated by the realization that one of the safety-enhancing measures proposed for the RBMK reactor design after the Chernobyl disaster was to raise the enrichment grade of the fuel, apparently to reduce the type's susceptibility for core poisoning and the resulting power fluctuations (at least that's how I understood it). I'm interested if the converse is also generally true, that is, whether it would be hazardous to let the fuel deplete too much in modern PWR/BWR reactors that don't share the design flaws of the RBMK.