Contains Rainbow Rocks spoilers
Hmm..I re-watched the film and figured out 12 of the 16 bands whose icons appeared on the tourament chart near the beginning of the Dazzlings' "Under Our Spell" I don't have a picture yet, but I'll list the official names and unofficial names of the bands.
Rainbooms: Our Humane 6 (icon: horseshoe and rainbow)
Dazzlings: Our Villains (icon: musical notes and red siren pendants)
Flash Sentry's band (icon: a speaker that looks like Flash's cutie mark)
Trixie and the Illusions (icon: moon and stars like the one on Trixie's wizard hat)
Snips and Snails' rap (icon: brick wall with graffti of a pair of scissors and a snail)
Crusaders (icon: 3 skulls on fire; fire matches the girls' hair colors)
Bulk Biceps and his violin (icon: Weight and wings; resembles B.B's cutie mark)
Octavia and her Cello (icon: pink treble cleft surround by cello bows; resembles her cutie mark)
Derpy and her saw (icon: a gray, sour note and a muffin)
Lyra and Bon Bon's Piano Duet (icon: A Lyre harp on a set of piano keys on a pile of Bon Bon candies)
Photo Finish's band (icon: a magenta ribbon with a camera shutter in the center)
The Eco-Teens (icon: globe with recycle arrows around it)
Those are the ones I know for sure, but the other icons are of:
A Diamond surrounded by a spike collar like you see on a dog...Oh The Diamond Dogs! or at least the human versions of them!
A happy and sad face like you'd see in theater, so I guess the Dramas-Teens!
A green oval with a musical note shooting across like a comet..I don't know for sure, but they could be the techies? Yeah, it's green like a circuit board.
A lightening bolt with a skull, 3 gemstones of purple, blue, and green, and drumsticks...I have no idea in the slightest, but it's the last remaining band on the chart.
Hmmm...Looking at the flowchart of the bands before and after the montage, I conclude that were 4 official rounds that are shown of that chart. No, the round where the Rainbooms sang "Shake Your Tail" and heard SnS Rap happened before the untouched flowchart was shown, so I see THAT round more like the qualifying round that determines which band faces off with which in the first match. I shall make a chart that list which band was beaten by which.
Round one (The most outter rows of bands on the chart):
1. Trixie beats "the band with lightening bolt, skull, gemstones, and drumsticks (The most unidentified band)"
2. Lyra and Bon Bon beat "the green oval with comet musical note" or "Techies"
3. Bulk Biceps beats Snips and Snails (Seen in montage)
4. Rainbooms beat Octavia (Seen in montage)
5. Photo Finish beats "The Diamond Dudes"
6. Dazzlings beat the Dramas
7. Flash Sentry beat the Eco-Teens (Seen in montage)
8. Crusaders beat Derpy
Round Two (the 2nd most outter rows):
1. Trixie beat Lyra and Bon Bon (Seen in montage)
2. Rainbooms beat Bulk Biceps
3. Dazzlings beat Photo Finish
4. Flash Sentry beat The Crusaders (We don't directly see the battle, but the Crusaders flash onscreen after the Trixie vs. Lyra and Bon Bon match, so I guess that was the Crusaders vs. Flash Sentry)
Round Three (Semi-Finals):
1. Dazzlings beat Flash Sentry
2. Despite the Rainbooms not finishing due to "Sunset going back to bad girl", the Dazzlings manipulate the Principles to officially declare the Rainbooms the winner over Trixie as part of their plan.
Round Four (Final):
Due to a "tricked Trixie trapping" (say that 3 times fast)our heroes below the stage, Trixie goes onto the Finals; the Dazzlings not so much wanting to officially win the BotB as much as create complete negativity and control over the whole student body, even sarcastily say "I don't think we can top that!" The Dazzlings get their full power and take One-Winged Angel Form. Nothing can stop them now!
The True Battle of the Bands (battle that determines the fate of the Equestria Girls World):
The Rainbooms, with sound system support by Vinyl Scratch, sing Fluttershy's song and take Anthro form. The Dazzlings overpower the Rainbooms with astral projections of their true forms. Sunset finally takes a stand and sings! She too takes anthro form. The student body is free and join in the chorus! Our heroes summon an AstralAlicorn--an Astralicorn, if you will--and finally defeat the Dazzlings, at last!