Just wondering what language to use in game menus, buttons, and settings.
The answer is probably you'd better prepare 2 versions of translation for the game menus/buttons to better sell your game.
However, either Simplified Chinese or Traditional Chinese is OK, since most of the case you could use a locale converter such as AppLocale under windows to convert different encodes. But if the game cannot use such tools, better prepare 2 versions.
As for the Classical Chinese, it depends on your game type, if it's an RPG, maybe some of the poem/idiom can be translated to Classical Chinese.
If you are asking about China only, then it is Simplified Chinese(SC).
All of main-land China us SC, except Hong Kong, which use Traditional Chinese(TC).
Taiwan (if you consider it part of China now) also use TC.
2011 Population (Worldbank.org):
China 1344M HK 7M Taiwan 23M