This was a tough decision. We have multiple rivalries, each for different reasons, and each has its own flavor. Each also has its own historical setting. For me, as a Vol fan since the early '60s, I would say:
Biggest Rivalry - Georgia now, because they are in-division and it has become a rivalry over the most important recruiting ground. Twenty years ago it would have been Florida for the same reasons.
Best Rivalry - Alabama (for those of us 60 or older, who remember) was always the classy rivalry, beginning in Bear Bryant's days, before hype videos replaced crying towels and poor-mouthing your own team as the ritual of game week. The game itself was a war from kickoff to final gun, but as soon as the game was over, there was mutual respect between warriors, and earnest celebration was saved for the locker room.
Ugliest Rivalry - Florida. The stage was set when Florida upset us in the '69 Gator Bowl and then hired away our head coach Doug Dickey. With SEC Division alignment, games against Spurrier's Florida were never classy the way Alabama's used to be. This was a hate-filled game for almost everyone, and not helped that everything was on the table in usually the second game of the season: recruiting the state with the fastest, largest talent... the SEC division championship... and in those glory years, the National Championship was just as much on the line.
Little (but smarter) Brother Rivalry - Vanderbilt. I never got why this was such an emotional rivalry for Vol fans, until I lived in Knoxville for two years. There really seems to be an anti-education reaction involved, as Knoxville is far more Appalachian in culture than the Chattanooga I grew up in. But for someone age 110, Vandy was a straight-up bare knuckle gridiron rivalry!
Little Sister Rivalry - Memphis is just a different place from the rest of the state. It's similar to the difference between New Orleans and Louisiana north of Lake Pontchartrain. Memphis sports teams could stand on their own accomplishments with deserved pride, but it seems like there's this underlying resentment that they're somehow excluded by their differences, that they'll never just be "one of the boys" around the state-wide Thanksgiving table. They don't want to fit in with the rest of Tennessee, but they do want us to wish they did.