The European version of Super Paper Mario contains a glitch where if the player enters all unlocked rooms in Chapter 2-2 and returns to Mimi without getting the key to progress to the final, locked room, the game will freeze upon talking to Mimi (note the movement of the flying Pixls), and the console needs to be restarted. Nintendo recalled many copies of the game upon discovering the glitch after release to replace them with an updated version, so this glitch can mostly be found in Super Paper Mario discs purchased in 2007.
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Unused enemy found in the code of Super Paper Mario that appears as a circle of four rapidly rotating Marios. The Marios are in fact copies of the playable Mario - they mimic all of his animations, and attempting to Tattle them gives the same information as Tattling the playable Mario. Jumping onto the Marios hurts Mario - although whether this is due to the circle’s status as an enemy or due to Mario’s jumping hurting the rotating Marios, hurting him in turn since they are copies of him, is unclear.
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In Chapter 3-1 of Super Paper Mario, entering a certain room causes a cutscene to start automatically where a Koopa Troopa panics upon seeing Mario, hits a Question Block, and collects a Mega Star from it. However, pressing the Home button on the Wii Remote to activate the Wii Home Menu directly after the Mega Star is released will cause the Mega Star to change its trajectory and remain stuck bouncing between blocks. This results in the Mega Star never reaching the Koopa Troopa, locking the game in this state.*
*Note that it is possible to reverse this by repeatedly activating the Home Menu again, although the timing on this is very precise and may take many tries.
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One of the bonus rooms in Chapter 7-1 of Super Paper Mario contains a pipe that releases Squog enemies. If the Pixl Boomer is placed onto the pipe and detonated at the moment a new Squog is released, the Squog, instead of being defeated, will flip upside-down and continue moving backwards. It will also shoot projectiles out of its back side.
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In Super Paper Mario, it is possible to perform an infinite jump by utilizing the following glitch: With Thudley active, jump and stomp in midair. Stomp again as Mario starts to move down. This allows you to switch into 3D mode in midair. Once you do, Mario can jump again, and the trick can be repeated indefinitely. This can, among other places, be used to skip almost the entirety of Chapter 7-4 by jumping over the gap intended to be traversed by building the rainbow bridge. (Source)

In Super Paper Mario, a speech bubble’s poorly-aligned tail makes it appear as though Peach is saying “HOT BABE IN ROOM!”. Francis’s speech bubbles are coded to come out of the resting position of his mouth, but as he throws his body backwards during this animation, his speech bubble seems to come out of nowhere. In addition, by coincidence, the bubble’s tail lines up with Peach’s face, resulting in this effect.

The South Korean designs for the Super Mario Galaxy (called Super Mario Wii in that region) and Super Paper Mario discs. Both discs feature original designs not used in any other region.

In Super Paper Mario, the exterior of Fort Francis is visible for only a few frames in full during a cutscene at the beginning of Chapter 3-4. After a fraction of a second, Bowser begins talking and his speech bubble obscures the top half of the fort, after which the camera focuses on the characters and never returns to the wide shot of the area. In addition, it is impossible to watch this cutscene again on the same save file. I have taken an enhanced screenshot of the fort during the brief moment it is visible, at four times the game’s normal resolution. This should be the most high-quality image of Fort Francis possible.

In Super Paper Mario, two identical-looking Flipside citizens that can be found on the 1st and 2nd floor of Flipside are both named “Muffy”. Given how they have names that are different from each other in all other languages, this seems to be an oversight on part of the Nintendo of America localization team.