I don't wanna get killed by Ku's fans, but... Rivalry? That's ridicilous. They are totally different just as their music styles, in fact - Kumi is after sexier image, she's more like a body-seller now, even though her music has improved a lot. She doesn't writes her own lyrics & music, etc. Ayumi is just pure beauty & splendid singer. She is the most talented singer I even heard about. She does write all of her amazing and such meaningful lyrics just by herself and she composes some of the most enjoyable melodies

Despite that I was a big Kumi fan (she was my top artist for a long time, just before Ayu) and she was my very first japanese artist, I'm with Ayu.
Ayu's the BEST
I think it'd be fun to make a duet with Ku