Ayu has so many wonderful performances, but a few of my favorites would be (in no particular order):
A Song for ×× @DOME TOUR 2001 - my absolute favorite, I love this one so much.
A Song for ×× @a-nation 2008
A Song for ×× @A Museum
Dearest @A Museum
winding road @COUNTDOWN LIVE 2004-2005
A song is born @STADIUM TOUR 2002
A song is born @POWER OF MUSIC 2011
HEAVEN @ARENA TOUR 2006
M @POWER OF MUSIC 2011
End of the World @COUNTDOWN LIVE 2000-2001
INSPIRE @COUNTDOWN LIVE 2004-2005
Trauma @COUNTDOWN LIVE 2004-2005 - The funniest ever, I love it
There are A LOT of other performances I could list, but I'll live it like this.