My favorite competition category is probably time attack. I have an affinity and knack for ring attacks and boss attacks too, so the window between all three is pretty close. I like time attack the most as it is what Sonic is all about; going fast and trying to get the best time possible. It tests your skill as time attacking is usually longer than boss attacking and requires more patience, as the longer playtime for each level can result in more areas to make mistakes. I think time attacking is an overall tough category to compete in, but so are the others as well...
I enjoy ring attacks as it is fun manipulating the level and finding a lot of different strategies to collect rings, and it also gives you a break from trying to go as fast as possible. Ring attacks display your creativity, and I have a lot of fun doing them for various Sonic games (AS LONG AS THEY DON’T INVOLVE RANDOM/LUCK BASED CRITERIA!) :( Team Chaotix RAs and Sonic Adventure RAs are not fun to me; I don't like the idea of having random stuff determine whether or not I get a good score.
Boss attacks are really fun too. Finding the best strategies to defeat bosses is another thing that I like to do, as it also displays creativity. I like to play boss attacks for various Sonic games, and the differences from game to game is exciting, as new strategies and techniques are required to defeat those bosses as fast as possible.
One category I like to do but haven't really dabbled in too much is score attacking. I like the concept, but just don't play that mode that often. I plan to do so in the future more often though, because to be a great player, you need to be good at all phases of competition. I probably will try score attacking more sooner rather than later, but usually every Sonic game is different for the calculation and set-up of how to do score attacks. I'd have to spend a lot of time doing that for multiple games, and I have other phases of some Sonic games to deal with first.
Special stages are the least played category of mine; someday I will practice them more and get better at them.
So, I like all of the competition categories, but time attack is probably my favorite, with ring attack and boss attack tied for a close second.