Moon Cave Tricks
Explanation
The Moon Cave is one of the most broken dungeons in speedruns, with a variety of tricks and skips available in all categories.
Moon Cave Skip
In NG+ Any% and NG Any%, the Moon Cave is skipped entirely via City Checkpoint Skip. Inferno is never required in the remainder of the game, except in Wawku Shrine to melt some ice blocks. NG+ can skip Wawku via Ark Early, and NG obtains Fireburst Early to melt those blocks instead.
Moon Cave Early
In NG+ AB, it is possible to enter the Moon Cave without defeating Crimson Helm by KT'ing over a very high invisible wall in front of the cave, then falling into a loading zone placed behind the barrier which normally blocks the entrance. See https://youtu.be/IA6FxpIdxRM?t=3487. It isn't known why this loading zone exists, as the Moon Cave is normally entered via cutscene, and cannot be left or accessed again later.
Calcified Cavern Skip
In NG+ AB, you can immediately KT out-of-bounds from the Calcified Cavern to reach the loading zone to the main Moon Cave dungeon.
Demonstration: https://youtu.be/IA6FxpIdxRM?t=3554
Early Floor 3 Vine
After entering the main dungeon area, you are intended to go to the kitchen, rescue Ajimi, and use the key he gives you to open the lockjaw door to get the Ogre Liver. This can be skipped by simply jumping and Vining to the flower next to the Imp Merchant on floor 3. From there, you can jump down to floor 2 and enter the Ogre Liver section from the exit.
This is a notable oversight, since it doesn't even require double jump.
Demonstration: https://youtu.be/p43X6EVyFdM?list=PLqC5BKx5eR95mzsPwkwNf-MvKgoP0gcGN&t=386
Ogre Liver Cave Skips
On the way to the Ogre Liver, the first flower can be skipped by doing a gold dash jump and air tackle for max distance (see https://youtu.be/p43X6EVyFdM?t=435). The second-to-last flower can be skipped by simply Vining directly to the last one. While Ammy is being pulled by the last Vine, one can Power Slash the eye switch. Ammy continues moving during the cutscene, saving some time.
On the way back, the first flower can be skipped with a jump from the right edge of the platform (https://youtu.be/p43X6EVyFdM?t=507). After Vining to the next flower (the middle in line), Ammy can do a mid-air wall-jump to skip the second-to-last flower (https://youtu.be/p43X6EVyFdM?t=517).
The aerial wall-jump doesn't seem to work if one has double-jump, so NG AB takes an alternate route. On the way back from getting the Ogre Liver, you jump, double-jump, and air tackle off the right edge of the cliff, then Vine directly to the middle flower (https://youtu.be/1axZtxRQ-x4?t=9570), then Vine to the next two flowers normally.
NG AB can also use double-jump to skip the small cutscene that shows the Waterspout pillar pond on the way out of the cave. See https://youtu.be/O4pY_qZ6fec?t=9606.
Floor 4 Galestorm Skip
On floor 4, there is a gap which you are intended to cross by using Galestorm to raise some banner platforms. This can be skipped with a precise gold dash jump and air-tackle (see https://youtu.be/p43X6EVyFdM?t=589), or with double jump.
Sand Room Early
On floor 3, there is a side-room filled with sand which includes a ball-pushing puzzle. It is possible to clip directly into the end of this room from the cannon area on Floor 4. This involves a fairly precise jump, double-jump, and air-tackle-bounce to clear the collision on the wall. The ceiling has no collision, so Ammy goes straight through. The sand room is directly beneath the cannon area, so Ammy just falls right in.
Demonstration: https://youtu.be/p43X6EVyFdM?t=617
Sand Room Ball Switch Trick
One can skip the ball-pushing puzzle in the sand room by drawing a CB next to the switch and using Galestorm to roll the CB over the switch. If done correctly, the cutscene of the switch being triggered will play, but the CB will roll off the switch before the cutscene finishes, causing all of the torches in the dungeon to stay lit. If the switch is pressed again after this, even for an instant, the torches will become disabled again if the switch is released. For this reason, most runners will immediately PS the CB to destroy it and prevent it from rolling back onto the switch.
If this fails, as a backup, one can go to the exit door of the room, staying near it to keep it open, and draw a CB. With a careful position and camera angle, it's possible to draw a Waterspout from the bowl of water on floor 2 to the CB, extinguishing its fuse. The deactivated CB can then be pushed onto the switch to depress it indefinitely. See https://youtu.be/MTEK1ttIXM8?t=10192.
Inferno Early
Within the sand room, there is a part of the upper cave wall without collision. It is possible to clip out-of-bounds here by doing an Air Ground Tackle off one of a nearby sand pile, jumping, then either doing a wall-jump or a double jump and curving Ammy over the wall.
Wall-jump version: https://youtu.be/p43X6EVyFdM?t=745 Double jump version: https://youtu.be/1axZtxRQ-x4?t=9648
Quadruple Eye Switch Skip
On the way to Inferno is a door locked by four eye switches which must be Power Slashed simultaneously. It can be skipped by jumping through the non-solid collision above the door.
Wall-jump version: https://youtu.be/p43X6EVyFdM?t=814 Double jump version: https://youtu.be/1axZtxRQ-x4?t=9653
Bouncing Flower to Floor 4
It's possible to reach the first floor directly from the bouncing flower on Floor 1 which opens after getting Inferno. This is done by pressing the jump button just after the flower reaches the bottom of its up-and-down motion. See https://youtu.be/p43X6EVyFdM?t=1044.
Floor 4 Black Imp Fight Skip
At the entrance of the cave to the Black Demon Horn, there is a trigger for a fight with a single Black Imp. This can be skipped by jumping around it, hugging the cave walls. This can be done on both the left and right side of the cave, but it is more lenient on the right side (when going toward the Horn). The skip must be repeated going the other way when leaving the cave.
Demonstration: https://youtu.be/1axZtxRQ-x4?t=9825
BF1 Skip
It is possible to skip basement floor 1 by performing Inferno Early (or obtaining Fireburst) so that the Eye of Fire can be obtained on the first trip across the collapsing bridge on floor 3. Then, one can double jump around the trigger for the collapse sequence to avoid falling to BF1. This is thought to be impossible without double jump, so it's only done in NG AB.
Demonstration: https://youtu.be/1axZtxRQ-x4?t=9934