I don’t believe much new needs to be said about sixths: they are thirds, major or minor, with an octave transposition.   Because of the wider distance, some difficulties exist in performance.   Your voice might need to shift from chest to head, for instance, whereas in a melodic third, one would typically try to stay in the same voice.   Popular female vocalists very frequently use the change of voice register as a feature of their singing, so sixths are great there.

Piano tuners use the sixths, too, in preference to thirds, in low registers because the relevant beats are twice as fast (and the lowerer you go the slower all beats).  

This post is about major and minor sixths; I’ll get to augmented sixths in a while.