In versions 1.0 and 1.1, it's possible to get the Eyeball Frog from King Zora when he would normally give you the Zora Tunic. You'll need an item in the adult trade item slot (anything other than a blank slot works, even an Empty Bottle), an equipped Shield, and you cannot already have the Zora Tunic. After unfreezing King Zora, talk to him. Before the dialog ends, simply hold R to shield on the frame after the dialog is closed. There is no timing involved.
If you collect the Odd Mushroom from the chest in Gerudo Fortress as a Child and save and quit, it will revert back to Cojiro since the game interprets it as failing the timed trading sequence. If you show this Cojiro to the guy with the Odd Mushroom, he will trade for it.
If you put item #50, the Poacher's Saw, onto B and then talk to the Carpenter in Gerudo Valley, you can trade for the Broken Goron's Sword without having done any other steps in the Biggoron Sword side quest.
You can do this even if you've already traded all the way to Claim Check. If you go back to Biggoron, he will act like his eyes are in pain again. Showing him the the Broken Sword will say "This item does not work here!", and if you talk to him he will say that the "sworrrrd" is his "finest worrrrk". So you cannot complete the Biggoron quest a second time.
On N64 versions 1.0 and 1.2, you can obtain a Spooky Mask by opening the Gerudo RBA Chest. You can then sell it just like a normal Spooky Mask to unlock the Bunny Hood mask from the Happy Mask Shop early (though the Spooky Mask remains unlocked). This can be used to skip the first couple trades and get to Mask of Truth faster.
If you show Biggoron the Bomb Bag (20) (see Upgrade Oddities) when his eyes are itchy they will work properly as if they are the real eyedrops.