I know I shouldn't start with one of the absolute best moments, but former BYU nose guard Lenny Gomes threw down the greatest quote of all time in 1993, when he had this to say about the Utes:

“All those guys think that’s all there is to life. But when I’m making $50–60,000 a year, they’ll be pumping my
gas. They’re low-class losers.”

Thanks, Lenny, for the laughs that continue 16 years later...

Honorable Mention: 1928: BYU's first non-loss to Utah, a 0-0 tie.