The Dood

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.

Collection of all types of items retrievable from dedicated Yoshi's Island claw machines, available in Japan in 1996. Note that the two Baby Marios have different expressions and arm poses.

Source: blacktangent

In Mario Kart: Double Dash, the place where the drivers emerge from after going inside the Warp Pipe on the Peach Beach track is supposed to be an open window above the pipe (the only open window on the track, making it clear this was the design intent), but this is nearly impossible to see in-game.

Top: the closest this can be seen is in two-player mode if one player turns around to look at the building, but the game's camera does not move up high enough to show the window unless the second player is very far away. As such, this is basically not visible (note Mario and Luigi being barely larger than a pixel on Wario and Waluigi's screen). Also note that this game does not have a rear view button so Mario and Luigi cannot look behind them as they would be able to in later games.

Bottom: only with external tools can the camera be shifted in such a way as to actually show what is happening. Note that the window is aligned with the Item Boxes but not with the exit point of the drivers; this is possibly a remnant from earlier in development where the arc was different so that the drivers could emerge directly from the window and grab the Item Boxes.

Source: Peardian