After speaking with Instructor Horwell, save at the Upper Academy statue. Activate BiT via the killing post (the top right post on the bridge leading to the goddess statue). Perform a BiTSave using the Upper Academy statue to reach BiT Skyloft, then run to the goddess statue to obtain the Goddess Sword.
Note that, due to text differences, this trick is actually slower in the Japanese version by about 3 seconds. Thus, this trick would only be used for PAL or NTSC version speedruns.
After the flight tutorial, it's possible to skip the cutscene leading into the Wing Ceremony by sidehopping out of the trigger before it starts. Hold Z while you select cancel at the save prompt, and then mash or time the sidehop. The window for this is only 2 frames, so it can be difficult to get. Once you're outside the cutscene trigger, you can dive off Skyloft to start the Wing Ceremony.
As a backup strat, you can save at the prompt before attempting this trick; then if you miss it, activate BiT via a Corrupt File and BiTWarp to the Bazaar statue. You can then dive off the platform to start the Wing Ceremony. This loses a minute to the sidehop method, but saves time over watching the cutscene.
After obtaining the Sailcloth, save and quit at the Upper Academy statue. Activate BiT using a Corrupt File. Perform a BiTSave using the Upper Academy statue to reach BiT Skyloft, then run to the Goddess Statue to place the Emerald Tablet.
This is only a couple seconds faster than chasing Fi optimally; also note that if you do not do Early Goddess Sword, you will not have a Corrupt File, and thus must chase Fi normally.
When you first fly to the surface, jump off the platform at the bottom of the lower academy stairs. Curve your skydive so that Link is aiming to the left and then call your loftwing. You will hit a spawn point closer to Faron than if you had just jumped off normally.
After the flight tutorial, save and quit at the prompt. Activate BiT using a Corrupt File. Perform a BiTSave at the Upper Academy statue to reach BiT Skyloft, then run to the goddess statue and place the Emerald Tablet. Continue normally from there.
As an additional timesaver, you can perform modified BiTMagic. This allows you to bonk the tree near the bazaar to open the academy gates, skipping having to jump off Skyloft and respawn and saving a few seconds.
The Sailcloth Skip is only useful for Harp speedruns; it leads to a dead end in full game runs, as skipping the Sailcloth in this way also skips a trigger that unlocks Instructor Owlan's door in the Knight Academy. This makes you unable to learn the Spiral Charge after defeating Imprisoned 2.
This trick can be used in Harp speedruns. It is a dead end in full game runs, as you must use a bottle in multiple places.
It is possible to do the Wing Ceremony more than one time. After obtaining the Sailcloth, save at the Upper Academy statue. Activate BiT (a Corrupt File is probably fastest). Perform a BiTSave using the Upper Academy statue to reach BiT Skyloft. Enter the goddess statue and immediately leave. When you dive off Skyloft, the Wing Ceremony will start again.
After you win, you'll do the Sailcloth skydive again, but nothing will happen when you land. You can then dive off Skyloft to do the Wing Ceremony again, and repeat this as many times as you want.
To escape from this, perform another BiTSave to reach BiT Skyloft and place the Emerald Tablet.
It is possible to clip through the floor in the Practice Sword room of the Sparring Hall. Use a bomb to Highflip onto the bins of swords, then jump towards the crate. You should slowly land, and by moving around and using first person view, you can fall through the floor.
There is no known use for this trick, but it could lead to other clip ideas.
Enter Karane's room in the Knight Academy. Roll so that you clip partly into her desk chair, release the control stick, and mash A, and you should sit in the chair. When you stand up you'll end up on top of her desk. From there you can jump along her furniture to reach the wall next to Zelda's bed, where you can sleep. When you wake up, you can run into Zelda's room.
This trick is not useful for full game runs, because Zelda's door is locked and it's impossible to escape her room.