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In Super Mario Galaxy 2, crouching at the edge of a mud puddle will cause Mario (or Yoshi) to walk if the crouch button is continued to be held, although still at crouching speed (internally, he is in the crouching state, merely with a walking animation).

This results in "phantom mud" where Mario appears to be stepping into invisible mud even on clean surfaces as long as the button is held.

Source: SPG64

In Super Mario World, the secret Soda Lake level is a unique location in that it is the only place to feature the Torpedo Ted enemies. These enemies were evidently designed to only be used in this very specific scenario and as such do not have proper interactions with even some of the most common items programmed.

Due to the fact that there are no Super Stars in Soda Lake, it is normally not possible to see that Torpedo Teds were never given an intended interaction with it. By modifying the game to place a Super Star in the level, the bizarre unintended interactions can be seen.

Note how the Torpedo Teds don't actually "die" when hitting invincible Mario, and are merely knocked back in an arc, yet still continue moving. At the end of the footage, it is also shown that catching a Torpedo Ted right after it emerges allows it to be hit on sequential frames, quickly gaining enough points to get multiple 1-Ups.

Top: artwork used on a rare officially licensed 1986 Famicom game case, depicting the cast of Super Mario Bros. riding a car.

Bottom left: this artwork would later be used for the album cover of a 1986 Super Mario Bros. Japan-only vinyl record.

Bottom right: the place this artwork was most famously used was the cover of All Night Nippon: Super Mario Bros., a limited-edition retool of Super Mario Bros.: The Lost Levels for the Famicom Disk System that replaced some characters with caricatures of Japanese radio personalities.

With the bottom two in mind, looking at the top image reveals that due to their format, they have been cropping out details of the original drawing all along. In particular, the smiling sun on the left, the full fortress on the right, and most notably the Spiny riding a Lakitu's Cloud in the top right can not be seen in any of the later versions.

Spinies have never actually been able to ride Lakitu's Clouds until Super Mario Maker in 2015, 29 years after this image was drawn.

Source: bigafrodogg

Due to an oversight in the code in Super Mario Sunshine, whenever Mario dies while on a tightrope, his weight value becomes infinite. This results in him continuing to drag down the tightrope for the entirety of the death animation until the fadeout.

There is no limit to how far he will drag it down; if the tightrope is placed above a bottomless pit and the timer removed, he will fall until he underflows the coordinate system and the game crashes.

Source: bowserliker98

Here are the two most powerful attacks capable to be inflicted by enemies in Paper Mario: The Thousand-Year Door, both in terms of most damage in a single attack (top) and most total damage in a multi-hit attack (bottom)*.

Top: the highest single-hit damage can be dealt by Marilyn in the Chapter 8 battle, where she can deal 48 damage after being made Huge by Beldam, applying two Charges, and Mario wearing the P-Up D-Down and Double Pain badges.

Bottom: the highest damage in a multi-hit attack is 80, after 10 hits from Magnus von Grapple 2.0's Audience Cannon with Mario wearing the P-Up D-Down and Double Pain badges.

* Note that this does not include audience manipulation since with extensive enough manipulation it is possible to make Macho Grubba do arbitrarily high damage, though that would be nearly impossible without external tools.

Source: Koopatrol

Extremely rare officially licensed 1993 Super Mario World ball-throwing ticket redemption arcade game. The cabinet features many pieces of artwork from various sources, with some being edited, some recolored, and a few completely original drawings (please zoom in to view details).

Source: Rykitu, jus

Super Mario 64 contains homing 1-Up Mushrooms in some areas. These relentlessly chase Mario by flying straight towards him at all times, and can pass through walls, making them very difficult to purposefully avoid (which is the basis of many self-imposed challenges in this game, such as collecting all Red Coins in a level while avoiding the mushroom).

However, the 1-Up Mushrooms have one weakness: they cannot pass through out-of-bounds areas. Normally, neither can Mario, making this irrelevant; however, in the castle grounds, there is a famous glitch whereby Mario can clip through the tower and land in a room on the other side of the door, which is in fact divided from the playable area by a thin out-of-bounds strip.

The footage shows the mushroom's behavior if Mario manages to enter that area while it is chasing him: it will keep flying against the out-of-bounds area, fruitlessly trying to get to Mario.

Top: in Paper Mario: The Origami King, the moat around Peach's Castle does actually reflect the castle, but due to the narrow width of the moat and the angle at which it is seen in most scenes, very little of the reflection is actually visible. Note the slightly visible reflection of the towers near the bottom edge of the screenshots.

Bottom: due to so little of the castle being visible in the reflection, the designers did not properly mirror the castle for the reflection model, and instead merely rotated it 180 degrees, which can be seen here in a model viewer.

As such, the reflection has the stained glass Peach looking in the opposite direction (circled on the left), as well as having the turret near the center of the castle on the wrong side (circled on the right). Note that rotation is one of the easiest operations that can be performed on a 3D model so this was likely simply made as quickly as possible to still be good enough for the small part of the reflection seen in-game.

Source: noclip.website

In Super Mario Galaxy 2, Fluzzard is only intended to be used as regular Mario, not any of his powered-up forms. Since the Fluzzard object was never intended to be interacted with as Boo Mario, no routine handling it exists; yet surprisingly, instead of the game crashing as would be expected, it fails in a unique way.

Top: as seen during gameplay, Fluzzard's legs appear to extend far behind the camera when Boo Mario glides with him.

Middle: changing the camera angle reveals that as long as Boo Mario remains attached to Fluzzard, he will keep extending his legs. The talons are not stationary, but do lag very far behind Fluzzard's body, causing the legs to stretch.

Bottom: Zooming out after flying further (please zoom in to view). The longer Fluzzard flies, the longer his legs will become until it is nearly impossible to see them when zooming out due to them being stretched so thin. Note that this does not affect the functionality; it is still possible to make it to the end and win.

Source: HEYimHeroic