The following is the ruleset for individual level runs.

If the individual level is tied to an episode, which is not a contest, then all powerups/characters are allowed.

However, if it is tied to an episode such as Lowser's Conquest Beta and The Invasion 2, only the powerups/characters you can gain access to are allowed in speedruns of that game.

Without further ado, here are the legal powerups/characters for current games.

The Invasion 2: (Only the powerups/characters created before 1.3.0.1 are allowed) Characters: All Powerups: All Shoes: All Yoshis: All

I would like to state that the list of allowed Yoshi's for Lowser's Conquest is to be determined since there is a Green Yoshi in the episode... Just it's access requires 120 green stars which according to the creator, their are 120 Stars. (I only found 119). These rules have always been however.