This page details bugs of EarthBound Beginnings.
If you manage to get Ninten's dog far enough from your house, the dog will turn into a female NPC due to graphic bank swapping.
If you talk to Pippi (who's only in your party temporarily) after you escort her home, she'll give Ninten her inventory. If she has used the Bread Crumbs, but Ninten hasn't, using them now will warp you to coordinates (0,0) and you'll be able to walk pretty much anywhere.
One of its main uses is to bypass the rock sealing Giegue's lair, which doesn't normally open until you get the eight melodies, which is used in speedruns of the Japanese version.
This was fixed for Beginnings and Mother 1+2.
Near the end of the game, constantly using the Flea Bag item on R7038XX will drop his offense to the point where he cannot kill EVE, and after 255 turns the battle automatically ends (as per the game code). While the event for EVE dying and giving you the melody happens, she also remains in your party, letting you retain an obscenely overpowered team member.
Beginnings and Mother 1+2 fixed this so that the Flea Bag cannot be used on the indestructible robots.