Øystein Ore proved in $1951$ that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.
In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative.
Ore's theorem was proved in $2008,$ and relies on the classification theorem.
- See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, Journal of the European Mathematical Society, vol. $12$, issue $4, 2010.$ pp. $939-1008.$
Interestingly Ore himself was mainly concerned with graph theory.