I dont really follow competitive chess but Im thinking countries like USA, Britain or China would be leading the way in terms of success and holding competitions. Maybe even Singapore or Korea.

I just skimmed through but on the wikipedia page on chess many countries like Russia and Ukraine get a mention.

The former Soviet block countries are of course the primary source of Grandmasters and chess professionals generally. Nevertheless, with India and China's mammoth populations there is understandably an ever increasing amount of titled players coming from that part of the world as well.

Here in Australia (to my knowledge which may be a little out of date), we have one retired Grandmaster (Rogers), one active Grandmaster (Johanson), and one soon to be Grandmaster (Smerdon). There are also a few foreign nationals living in Australia who hold the Grandmaster title (Schmaltz), so the above only refers to citizens and permanent residents.

Correction, there are two active Australian Grandmasters: Johanson and Zhao. I think Russia is still definately the best country in chess.