A lot of Korean songs would fall under the category of K-Pop (kind of boy/girl band bubblegum pop, which is huge all over Asia, along with Korean movies and TV dramas). But most of the following are examples of Korean underground music.
Cut-out animation from Fin. This song keeps almost making me want to break out into Come On, Eileen.
Mono Diary. Street food, floating guitars, young love, church weddings. I can’t understand any of the words, but I think they’re pretty happy and nostalgic.
Nell – Stay. The intro to this video is a little more intense than the song itself.
3rd Line Butterfly. I rather like this one, on first listen.
And this is the song I’ve had in my head since I joined twitter (“follow me, follow me”). This one’s straight up K-Pop.
T-ara – Bo Peep Bo Peep
[partly via Indieful Rock]