Cheap Korean airlines ??

Does anyone have any suggestions on cheap carriers that fly out of Seoul (probably to Taipei,Taiwan)??

There are no budget airlines operating in and out of Korea at the moment except for Jin Air but they have very limited destination options. Depending on when you're travelling, Air Asia are commencing flights from Seoul in October, but that will probably be just to Kuala Lumpur.

Your best bet is to try directly with the airlines and also get in touch with some Korean travel agents (I usually use Shoestring Travel http://shoestring.kr/eng/

There is no 'cheap carrier' on S.Korea-Taiwan route.

In late September, for example, Thai Airways, Cathay Pacific, China Airlines would be your best bet, around KRW 320000~390000 for 14-day to 1-month return.

Look at EVA Air, they are the main Taiwan carrier, and fly direct to Seoul.