Whether you're good at singing or not, it doesn't matter. Karaoke places, known as noraebang in Korean, are everywhere. It's one of the greatest pastimes, especially at night. If you're wanting to drink and have some extra singing fun, it's perfect. For about roughly $10US (depending on the quality of the place) you and your buddies can sing up to 1 hour... and if the place isn't busy and you're bringing them in extra doe by buying goodies from them, they are bound to give you some extra singing time... even up to an hour extra!
Also, all the noraebang places have Korean, English, Japanese, and Chinese popular songs... so don't worry if you don't know any Korean, the English section has enough songs to keep you happy. Every month, the noraebang places update their archive with the most recent popular songs while also keeping the very old classics, like those by The Beatles.
Each noraebang room generally provides two mics and at least one tamborine or moracca, so if you don't really want to sing, you have those to keep you busy.
One place I highly suggest is called Luxury. The name states what it is. The place is beautifully designed, clean, and lit, and requires you to take off your shoes to keep the skid marks from off their glass floors. Also, they provide FREE unlimited ice cream. Self-serve, it says!