Non metal? Damn, I would be here all day, esp if I include all the punk shows.
Ok, I will just start listing non-metal bigger bands as they come to my head:
Neil Young
Sonic Youth
Sisters of Mercy
The Cure
Lollapalooza 92 (Red Hot Chili Poopers, Ministry, Soundgarden, Pearl Jam, Ice Cube, Jesus and Mary Chain, Lush, etc)
Lollapalooza 93 (Alice in Chains, Primus, Dinosaur Jr, Tool, Rage Against the Machine, etc)
Lollapalooza 94 (Smashing Pumpkins, Beastie Boys, George Clinton, the Breeders, etc)
Nine Inch Nails / Hole / Marilyn Manson
Morrissey
Public Enemy
Green Day
KISS
Ramones
Cheap Trick
David Lee Roth
B 52s
Depeche Mode
Grateful Dead
Front 242
Ministry
My Life With The Thrill Kill Kult
Bob Dylan
Goo Goo Dolls
RUSH
Def Leppard
Misfits
Social Distortion
Bad Religion
Offspring
Rancid
Warped Tour 97
Warped Tour 98
Warped Tour 99
Dropkick Murphys
Hatebreed
I am sure I am forgetting literally hundreds.
I am not even going to bother with listing the 200+ smaller punk shows I have been to, as the majority here probably won't know most of the bands.