This page attempts to document the bugs and glitches found in the game.
Exor glitch
Normally, bosses are immune against the instant death effect caused by correctly timing Geno's Geno Whirl. However, this appears to be not the case for Exor. Kill one of his eyes so his protection is gone, then use Geno Whirl on Exor and time it correctly and Exor will take 9999 damage and die instantly. This makes the boss fight extremely easy, and was probably not intended by the game developers.
Mute freeze
Peach has the Mute attack which normally casts Mute status on one enemy, but if you spin the Control Pad quickly, it will instead affect all enemies. In a case where you're fighting six enemies, trying to cast Mute status on all six enemies freezes the game. As there are very few battles with six enemies, causing this is rare - one such battle is found behind one of the Battle doors in Bowser's Keep, it features six Shamans.
Ultra Jump glitch
Ultra Jump can cause damage on enemies that are normally invulnerable to Jump attacks, and even enemies that are normally completely invincible. You just need one enemy that is vulnerable (not necessarily weak) against Jump attacks to attack it with Ultra Jump. As you jump repeatedly, Mario will also jump on other enemies including any invulnerable ones, and cause damage equal to the amount of times Mario jumped on the enemy. You can use this glitch, for example, to defeat the two Chef Tortes in the fight against Bundt, which will cause the game to freeze when one is trying to poke the other.
Over 9999 damage
It is possible to do more than 9999 damage. You will need an enemy that is weak against Jump and has Fear status (by using Bowser's Terrorize) and have the Super Suit and Attack Scarf equipped on Mario, as well as an Attack Up. Now, have him perform 100 Ultra Jumps on the enemy and it will do more than 9999 damage. However, as the game does not support a damage with more than four values, the first digit will be a green capital I.