First off it is not normal for plugs and outlets to start melting and "burnout". ^^^
If the breaker didn't pop that is concerning because that is exactly what it's purpose is.
So if 2 outlets are melting and the white wires were touching I would guess the outlets were wired wrong.
The wire runs from one outlet to the next one, the black wire to the black side of the next outlet.
If the black wire got installed to the white side of the next outlet nothing would happen until those 2 white wires touched because they arn't really both connected to the white side. (You get the point)
That breaker should have tripped though, I would check that and buy a couple new outlets and a tester.
Black wires to brass and check your whole house with a tester, 5 bucks.
http://www.ehow.com/how_5260048_test...-problems.html