One of the concerns about Chernobyl was in the event of a power grid failure, external power would not have been immediately available to run the plant's cooling water pumps...exactly what's happened in Japan. And why I suspect Merkel is drawing down those older nuclear power plants...make damn sure there's plenty of power available immediately without hesitation so they don't experience catastrophic failure like Chernobyl and Japan.

As to your question, the best I can figure out, cooling is the biggest problem so far :

3-mile island - On March 28, 1979, there was a cooling system malfunction that caused a partial melt-down of the reactor core.

Chernobyl - The disaster began during a systems test on 26 April 1986 at reactor number four of the Chernobyl plant, which is near the town of Pripyat. There was a sudden power output surge, and when an emergency shutdown was attempted, a more extreme spike in power output occurred, which led to a reactor vessel rupture and a series of explosions.

The wikipedia artile is more indepth...keep in mind this was during the cold war so info is at best ballpark variety

url - http://en.wikipedia.org/wiki/Chernobyl_disaster