What are some popular Korean songs?

I especially like ballads and trot.

There are a lot, I'll suggest singers not songs :)

TVXQ
Super Junior
FT Island
Paran
Tim
Sung Si Kyung
SNSD (Girls Generation)
Lee Seunggi
Eru
Kang Chul

and so much more.

Fly 鈥?Epik High
Love Love Love 鈥?Epik High
8:45 Heaven - Drunken Tiger

A lot more but I can't think right now. Kinda old, but still good..

Saranghago Inneyo
Neol Sarang Hagetso
Miracle

Wheesung is great, he is actually an R&B singer but not limited to that genre here are sample songs:

1. Sarangeun (Love Is) - a little old but still nice to listen to
2. Isangeul - same with 1
3. 雼れ硱霃?膦嬱晞 (Dachyodo Chohwa) - from recent album
4. Love Hero - from recent album

Fly to the Sky is also must listen duo
Lee Jee Hoon's Trinity is his most recent album but its worth listening although it was released 2004 or 2005 -- this one is pure ballad.

攵€霐?- V.o.s. this is my favorite song :)
觳皽甑Μ (green frog) - ITT :]
marry u - super junior
毵堨毵?鞚胳偓 (last farewell ) -big bang
and if ur lookin for trot, you should listen to super junior's trot album,
its called 搿滉靖旰?(rokkugo)

Dong Bang Shin Gi(tvxq)
Super Junior
F.T. Island
Fly to the Sky
Tim
Tei
V.O.S.
Big Bang
Paran
Wondergirls
Eru
Epik High
Lee Seung Gi and much more.... i suggested singers! and groups!

Daphine Loves Derby

Loveholic

for trot.....jang yoon jung 鞚措敯鞚措敯鞖?.
for ballads....like super junior kry, some dbsk songs ( ill be there is balladish), wheesung, fly to the sky, ft island, etc...
just type their name in and download every song lol..
i also like wheesung-靷瀾鞚€ 毵涭瀳鞏?its catchy
and wondergirls-tell me....is really popular, especially the dance.
ehhhh this is all i can think of this second...

my favorites that are often played on american radio:
fergie - big azns don't see or the remix big azns can't drive
geto boys - damn it feels good to be american