If you can't find a trash can in Korea, just start your own. Seriously. I've yet to start my own trash pile on the sidewalk/street, but I am guilty of adding my trash to someone else's pile.
These pictures are from downtown Seoul.
I think Korea is a great country, but in my opinion it does have a problem when it comes to trash. Not only is there a lack of trash cans on street corners and in subway stations, but lots of food items come individually packaged, creating more trash than is necessary. I come home almost every night to find some flyer or advertisement taped to my door, creating more trash. Walking down the street, I see these same flyers laying on the ground. I find myself laughing at my kids when they tell me that China is so dirty, because in my mind Korea isn't that much different.
Throwing away my apartment trash is rather complicated. I have to sort my trash into different bags - 1 special bag for food stuff, 1 bag for recycle materials, and 1 bag for everything else. When Huy (a new teacher) asked me tonight what I did earlier today, I told him I took out the trash. He gave me a strange look. He doesn't realize yet what a process taking out the trash can be.