As a long-time KCBS competitor, I can assure you that the vast majority for teams are cooking and turning in thighs in the Chicken category. Thighs have a great balance between muscle and fat which helps to produce a very good product if cooked properly. That having been said, drumsticks have become more popular in recent years, especially when given the Lollipop treatment. (Any search engine will lead to you multiple videos on how to make them.) The white meat options, breasts and wings, will often accompany a dark meat entry but are rarely used alone. The simple truth is that even the most succulent and well-cooked piece of white meat will often pale in comparison to a well-cooked piece of dark meat.
If you are a new competitor, I would recommend that you learn how to cook really good thighs and/or drums before pinning your competitive hopes on the white meat options. (Unless, of course, you are required to turn in white meat.)