If you find any glitches or time saves, useful or not, post 'em here!
Glitches:
Gliding: As you are running, you can glide down off a platform by pressing back and forward near a ledge. You will maintain good speed and control. If you double tap as you're gliding, you will break into a sprint when you hit the ground.
4-6 Air Stream Glitch #1: The first air stream glitch can be done right below the bonus level. You just let the airstream hit you against the roof next to the icicle. There are two entry points, and the one closer to the icicle is great for 100% because you can get the bell and the bonus level without buying a fire bird. The entry point farther away is not as useful, but can be good if you want to use Amy to collect the first bell.
4-6 Air Stream Glitch #2: This is by the second bell. Let the air stream take you into the wall next to Amy's tunnel and you can walk around in there. Haven't found any uses for it yet.
4-8 Wall Glitch: Using Shinji to fight the boss, you can get trapped between some of the stones blocking your path of entry. Use this to line up a one cycle kill. You should stay on the wall and slide along it.
Golden Bat Speed Glitch: If the golden bat is trying to collect moving coins, it may not be able to catch them and will follow them around in a circle. This causes the game to run much faster. The only level I've seen this in is 5-3. Haven't found any uses for this yet.
Water Jumping: I'm pretty sure the developers intended for Shinji to be the only cat that can jump off of water while holding down sprint. Haven't found any uses for this yet.
Opposite Ledge Jump: On an ice platform, if you are dazed by an enemy and are sliding towards another without control, a well timed sprint can have you jump off the ledge in the opposite direction.
Lava Jump: If you aren't dazed already, there's a chance that you can jump out of lava dazed and land on all fours somewhere safe. I'm not certain how to ensure this save every time.
Time Saves:
5-4 Butterfly Glide: Playing as any character, you can collect the butterflies and grab a bell right before the lava geyser blocks your path. Just quickly collect blue, green, pink, yellow. Then sprint and glide from yellow to white and begin a sprint on the way down. You should make it past the geyser in time!
6-7 Lever Jump: Playing as Kuro in the room with two levers, you can jump over the levers by sliding down the adjacent wall and jumping at the right moment. Thus, you can avoid flipping one while quickly getting to the other lever.
Lava Pause: Right before taking a death from lava, you can pause and exit to the menu.
UppercaseT found a glitch called Menu Storage in which you can hold down the cat switch menu and later release it to switch to a cat off of the switch pad. Doing so while in water saves the state of the original cat (Brutus), so next time you switch to Brutus, he thinks he's getting out of water and jumps. UppercaseT calls this Jump Storage -- this is the first practical use for it. In 2-7, I figured you could first run and jump with Alex and switch to Brutus in midair to get on the blue platforms. Instead, I discovered that the ceiling has no boundary. Thanks to this new strat and the finding earlier this week (oob to goal) I have officially obliterated 2-7 lol Oh and, unrecorded, I found you can oob after getting the bell and climb up the blocks instead of smashing them to get to the other side. So Shinji meets Shinji in a run is doable with only a few seconds time loss if Menu Glitch is executed quickly @MrUppercaseT :) Also, just confirmed Jump Storage works with lava. There could be some nice uses for it in lava based levels. For instance, we now know Jump Storage can work to skip the shop in 2-H and 5-H, but its not a time save.
Got in the mood to do some more skip hunting, here's what I've found so far. Credit goes to @MrUppercaseT for the very useful backwards ledge jump.
2-3 Shop Skip using backwards ledge jump: https://media.discordapp.net/attachments/409020609880653827/742933700957831278/Screenshot_20200811-213244.jpg?width=849&height=406
6-6 as Alex and Brutus only using menu storage:
6-7 with backwards ledge jump (All credit to MrUppercaseT for this one, I'm just copying what he found)
3-3 Out of Bounds Bell Collection with backwards ledge jump:
2-7 Shinji meets Shinji confirmed (can climb up the rocks out of bounds):
3-7 Switch skip with a backwards ledge jump:
4-7 Out of Bounds with Amy -> Shinji menu storage:
6-7 Switch skip using a backwards ledge jump. Enables full game runners to use Olly starting at 5-8 if they so desire :)
6-4 Alternate Wall Climb Skip Discovery -- summary after an hour or two of testing
It doesn't work yet. Why? Because you have to load the next part of the zone by running somewhere past Brutus's wall. However, once you run past Brutus's wall, the only way to get up to the key is with a jump storage, so you can't use jump storage again to clip into the ceiling.
Also tested: Backwards ledge jump with Alex gets him most of the way up the wall, but there's no way to switch him to Olly there.