98 Elantra - 145K. Vehicle died in the Canadian Tire parking lot. No charging light ever came on, but it is like the description above - lost all 12V systems one by one, then the engine.
I suspected the alternator myself, but since it was dead at Cdn Tire, I let them do a diagnostic rather than being towed to the dealer. They confirmed the alternator, but wanted $700 to change it. I bought a rebuilt unit and changed it myself. (Unfortunately, I had to give my old one as a core, and it appears there was nothing wrong with it. Should have bench-tested it.)
After the change, the problem persisted. Chasing it around with a multimeter, we found that the alternator would charge intermittently. It would sail along at 14.5V for 2-3 minutes, then quit, then it might or might not pick up again later.
Interestingly, the battery light on the dash will either work or not together with the alternator. If it's in a charging mood, that light will turn on at startup. If not, no light. The failure to charge seems to take the indicator light with it.
We are now chasing the problem along one of the two wires off the regulator - the one that governs the idiot light. Apparently, the behaviour of the circuit is that voltage on this wire governs the regulator's ability to charge or not charge. (I am paraphrasing here - my dad is the electrical engineer - not me.) When the battery reaches the desired voltage, the circuit switches off. If you lose a connection at this wire once that circuit is off, the regulator is unable to resume sending power to the battery.
The problem is likely not in the fusible link. It is somewhere in that wire. We need to chase it through the harness.
I suspect the battery is now damaged as well.