I used to program a KA-10 (one of the main primary research architectures of the ARPANET -- ask yourself why Berkley sockets work the way they do). Anyway it had core memory except for the machine registers which were semiconductors (DTL, not TTL). Those 16 machine registers were also the first 16 addresses in memory BTW.
Aaaanyway, the power in Cambridge would sometimes go out, and when it did whatever process was running would die, as all the register state (including PC which was a separate register) would be lost. But otherwise you could just restart since the core was fine. Of course if the monitor (i.e. kernel in today's parlance) was running, well you were sunk.
Core memory isn't wiped on loss of power, so you could read the contents out of a dead machine, to see the last state.