Blue October - Hate Me
Plenty of current artists and bands will stand the test of time, and a large amount of them have older roots to them giving a classic sound.
From The Black Keys (Tighten Up, Lonely Boy), Jack White/The White Stripes (Seven Nation Army, Icky Thump) to solo artists like Ray Lamontagne (Trouble, You are the Best Thing), John Mayer (Waiting on the World to Change, Gravity).
Newer catchy stuff with some substance will stick around, just like all big pop songs do. Bands like FUN (We Are Young) and Foster the People (Pumped Up Kicks) have songs that will certainly be heard years from now, wether you like them or not.
It will also have a lot to do with taste. I can't stand 95% of rap music, but it is popular. Wether I like it or not, I know the songs. I like late 90s and early 2000s rap, and hearing stuff like Forgot About Dre still doesn't get old for me. Rap with good hooks and beats like Kanye West and Outkast will last for some time.