Biggoron sword

You'll need Epona for the Biggoron Sword side-quest. Go to Ingo's ranch at night and learn Epona's Song from Malon. Use Epona's Song to practice with Epona during the day. Talk with Ingo on horseback to race him. Win the race twice and you'll get Epona.

Talk to the Cucco Lady in Kakariko Village and she'll give you the Pocket Cucco. Incubate it, then go into the house where Talon is sleeping and use it. Give it back to the Cucco Lady.
Go into the Lost Woods in Kokiri Forest and go left. Use the blue cucco on the carpenter's son.
Go back to Kakariko Village and go to the old lady's house at the back (near the windmill). Give her the mushroom.
Go back to the carpenter's son and give the Poacher's Saw to the girl.
Head to the Gerudo desert, jump the bridge with Epona and give the saw to the head carpenter.
Head to the big Goron at the top of Death Mountain. He'll send you to King Zora in Zora's Domain. You'll need to go inside the Ice Cavern dungeon (jump across the ice to reach the entrance) and get some blue fire in a bottle to unfreeze King Zora.
He'll send you to the researcher at Lake Hylia. The researcher will then send you back to the big Goron. Wait for several days and the sword will be finished.