I grew up listening mainly to "mainstream" Punk (Bad Religion, NOFX, Social Distortion, The Offspring, Pennywise, The Bouncing Souls, Dropkick Murphys- who were an oi band with a skinhead following when I was a kid, The Ramones, Iggy & The Stooges, No Use for a Name, Lagwagon, Guttermouth, Frenzal Rhomb, Green Day- Pre Dookie, Blink 182- Pre Dude Ranch, The ******, & The Living End) & "skacore" (Operation ivy, The Mighty Mighty Bosstones, Less Than Jake, The Mad Caddies, Reel Big Fish, Goldfinger, The Pietasters, The Specials, & Sublime).
I also listened to grunge rock (Nirvana, Weezer, Beck, Sonic Youth, Stone Temple Pilots, ******** Surfers, The Meat Puppets, The Violent Femmes, & Pearl Jam- before they started taking themselves way too seriously. I do kinda like them again though, as Eddie Vedder's soundtrack for Into the Wild is awesome & he's seemed to develop a sense of humor once again.).
And my other big influence was my Dad, who was into Dylan, Genesis, Phil Collins, Tom Petty, The Beatles, The Stones, & The Traveling Willburys (Who may just be the best band of all time).
I continue to listen to all of the above, but am also really into The White Stripes (RIP), The Raconteurs, The Black Lips, The Black Keys, the Ting Tings, Arcade Fire, Rise Against, Gorillaz (Who's new album would have sucked sans Mos Def), The Kings of Leon, The Lawrence Arms, Mumford & Sons- what they do without drums is pretty awesome, Tool, Vampire Weekend, & Wolfmother.
Most of the above list has made ****** modern music, but has good old stuff with a few exceptions: Green Day's last two albums were good, NOFX is still amazing, Bad Religion's still got it, The Offspring have some good new stuff, Weezer is still Weezer, STP's new stuff is pretty great, Beck is still awesome (I especially liked the Scott Pilgrim soundtrack), & both Tom Petty & the Stones still got it.
Bob Dylan is just phoning it in now, though.
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"The Rich are there to take all of the money & pay none of the taxes, the middle class is there to do all the work and pay all the taxes, and the poor are there to scare the crap out of the middle class." -George Carlin