Hyrule Castle

Barrier 1

With Bomb Arrows

If you have bomb arrows, you can have the bokoblins blow themselves up on a bomb arrow. Don't actually shoot the bomb arrow. Just hold it until they hit you.

With Bombs

If you have bombs you can kill all the bokoblins in one bomb. Just pull a bomb either right before or right after the barrier cutscene, turn around, and have the bokoblins hit the bomb on the ground. They'll then crowd around you and all get blown up by the bomb.

With an LJA

Without bombs, the fastest way to skip the first barrier is to do an LJA from the nearby stairs onto the hedge. Line Link up so that the edge of the third stair from the top is in between his legs, back up slightly, then aim above the top right corner of the hedge (which is slightly to the left of the edge of the hedge). Target twice to give yourself enough time to pull out the sword and LJA.

With a Moon Boots Finishing Blow

Only slightly slower is the moon boots finishing blow method. Bait a bokoblin into the corner of the hedge, quickspin through the hedge to hit it, then perform a moon boots finishing blow (do a finishing blow with iron boots on, then force unequip the iron boots within a few frames of starting the jump) to clear the hedge. Ideally, you'll land on top of the bokoblin or get hit by an enemy to skip the animation that plays when Link's sword is stuck in the ground.

With a Midna Charge

The Midna Charge method is slightly slower, but a bit easier. Just wait until all of the bokoblins have spawned and do a Midna charge. Most of the bokoblins should die, but you can wipe out any stragglers by just doing a quick second Midna charge. (Transforming back into human does not waste time if you do it after rolling out of the second barrier in the next room, as the timer for opening the gate will continue to run while Link transforms.)

With the Ball and Chain

Alternatively, you can kill the bokoblins by slashing each of them three times and then hitting them with the ball and chain. Slash the first two bokoblins three times, roll backwards a bit, slash the second set of bokoblins three times, walk forward slightly, then throw the ball and chain.

With a Box

If you don't want to go fast, you can also skip the first barrier by using a box to clip into the wall in the corner. Just drop the box by pulling out the sword as Link is stepping forward into the corner, then walk or roll through the seam in between the hedge and the wall.

Skip Barrier 2

Simply roll into the trigger hugging the fence on your right, skip the cutscene immediately and then hold down-right and roll, you will be outside the barrier. On the way back you can just walk inside the barrier wall as it is only one sided.

Skip Barrier 3

Simply Jump-attack towards the pillar on the right and mash pause and hold right. You will skip the cutscene as soon as possible and because link is in the air he will not walk into the barrier and holding right will keep you outside.

First Darknut Room

Strat by SVA161620

For the first phase, roll towards the darknut while pulling out the sword, rollstab, and then slash four times. After that, do a walking slash to make the darknut block and slash six times to take off the rest of its armor. Walk towards the darknut and sidehop right through the sword throw. Then wait for the darknut to pull out its second sword. Slash twice and use the ball and chain to finish it off with three hits (throwing the ball and chain for the third hit is often helpful). To get the perfect boomerang angle, stand inside the diamond on the floor and aim towards the pillar. When the cutscene ends, throw the boomerang and it will hit the torch.

Dynalfos With Bomb

Once you skip the cutscene pull out a bomb and run towards the Dynalfos, when they notice you stutter for a while and then throw the bomb at them. They will run away from the bomb when it's all the florr after a while hence why you wait for a bit.

Alternatively you can do a much safer and consistent strat which is barely slower.
Therefore just lure them to you and pull a bomb as soon as they start running towards you. When they are close, slash the bomb.

Aeralfos Fight with Ball & Chain

Discovered by Dragonbane and Garm

Before getting to the Small Key chest you have to fight Aeralfos. The fastest way to do this includes using the Ball&Chain. Enter the Barrier and skip the cutscene as soon as possible. Hold L and mash the Boomerang button to make Aeralfos shield and instantly clawshot him down.
While the clawshot is in the air, equip B&C over the boomerang and swing it. After you hit Arealfos 5 times, throw it in his direction - it will kill Aeralfos on its way back if done correctly.

Block Room Shortcut

As long as you don't stand on unstable blocks for too long you can just cut right across the room. To do this roll across and show in the video making sure you go diagonally so you don't stay on unstable blocks.

Last Group of Lizafos (Bomb-Arrow)

This only works for the last group of Lizafos as they are closer together than the group below. Before spawning the barrier simply bomb-arrow the wall next to the right Lizafos and it will kill both of them making the barrier disappear before you watched the cutscene to spawn it.

Skip the Final Darknut Fight

Discovered by DF6780

It is possible to skip the final Darknut fight by clawshotting the lantern right next to the boss door in Hyrule Castle. Get off the spinner early and position yourself carefully right before the barrier trigger. Clawshot the lantern and open the door.