I've always thought of sixths as dreamy, mellow chords - probably because the first one I ever picked up on was in "Band On The Run" where it alternates with a Dmaj7 chord. However, it's also a good chord to finish a song with - the Beatles did it with "She Loves You" and Rod & The Faces did it with "Cindy Incidentally," which also uses a 6th chord in the main riff - A6 alternating with Em, resolving to a D chord.
There is another use of them, though - as a powerchord in the blues/rock shuffle.
A5 - 5 7 x x x x
A6 - 5 9 x x x x
A7 - 5 10 x x x x
I wonder how many 6th chords Francis Rossi and Rick Parfitt have used over the past 40 years or so!
"Sometimes the beauty of music can help us all find strength to deal with all the curves life can throw us." (D. Hodge.)