I believe it was Ming that ran a tourney. Then I had an "unofficial" monthly game for a few months. Now there is Apolyton University (AU 106 is the latest). However, AU is not a competition, only a learning experience.
The most popular single player tourney going is on Civfanatics:
http://www.civfanatics.com/civ3gotm/
The Civfanatics August game had 143 entries and probably a similar number that played but did not send in games. Compare this to seven entries, that I got for the last Apolyton game I hosted.
If you do a search you can find a few more monthly tourney games, but many of them have gone the way of the Apolyton game (retired, faded away...).
- Bill