Japan: Ganbare Goemon: Neo Momoyama Bakufu no Odori (NTSC-J) 
NA: Mystical Ninja Starring Goemon (NTSC-U) 
EU: Mystical Ninja Starring Goemon (PAL) 
Currently, the NTSC-J version is the fastest version for all categories.
Speedrun-relevant version differences in MNSG:
Press and hold the C-Right button during a loading zone to despawn all objects, enemies, and cutscenes/events. Releasing the C-Right button will return everything to normal, except in certain locations where the game may softlock. This is only possible in the japanese release of the game and got fixed in the international releases.
Chain Pipe Skip
If you take damage in a corner, you will be slightly boosted upwards, allowing you to gain some height. However, this is not possible in the international releases.
In MNSG you can jump in the water, however if you are in corner and press the jump button a bit longer then just tapping it (not holding the A Button), you will jump a bit higher, wich makes some skips possible. This trick is NTSC-J exclusive and got patched out in the international releases.
If you clip into an object and slash on a specific timing you character will fly trough the map with a very high amount of speed. Mokkori slash duration depends on the selected character and objects.
In the Japanese version, you can beat the Mermaid Minigame by simply swimming left and quickly mashing up the waterfall. However, in the international releases, the objects in the river fall randomly, making it more difficult.