If we're talking Rushmore's mine would probably be Zeppelin, Daft Punk, Radiohead and... geez, the first 3 were easy, the next one could be a revolving door of bands and groups. Gorillaz? M83? Arcade Fire? Pink Floyd? Interpol? Jay-Z? Arctic Monkeys?
If we're talking genres, I'll readily listen to just about anything from rap to classical, but my favorites are probably:
Classic rock (Beatles, Pink Floyd, Bowie, E.L.O., Stones, Queen, Led Zep!)
'80's rock (The Clash, Tears for Fears, The Smiths, Stone Roses, G 'n R, Echo and the Bunnymen)
'90's alternative (Nirvana, Rage, Weezer, Pumpkins, Radiohead!)
Techno/electronica/edm whatever you want to call it through the years (Fatboy, Chemical Bros, Prodigy, Underworld, M83, Justice, DAFT PUNK!, Disclosure, LCD Soundsystem, Zhu, Flume, Bassnectar, Gorillaz, Empire of the Sun, Grimes)
Older rap (Big L, Biggie, NaS, Jay-Z, Eminem, Wu-Tang, Canibus, Danger Doom, OutKast, Tribe, Jada, Beasties)
And over the past 10-15 years, music that would fall into the indie rock genre and sounds that blur those lines (Broken Social Scene, Metric, Vampire Weekend, Phantogram, MGMT, Edward Sharpe, Foster the People, Awolnation, Naked and Famous, Black Keys, Arctic Monkeys, Interpol, The National, Doves, Editors, Alabama Shakes, Shins, Arcade Fire, Fleet Foxes, Dr. Dog, Grouplove, Muse, Delta Spirit, Local Natives, The Dig)
I just wish I had more time for finding and listening to new music these days, so keep the suggestions coming.