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Horizontal climbing is super slow, you want to jump as much as you can. Vertical climbing isn't helped by jumping. However, jumping off of a vertical rope gives you tremendous horizontal speed until you land; bouncing off of enemies maintains this momentum. If you jump from a vertical rope and land grabbing onto a horizontal rope, you can maintain that speed if you buffer a jump to go off as soon as you grab. This can be repeated with further buffer jumps to cross horizontal ropes very quickly, but is only useful a handful of times in the run, especially Ice Prison.
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If you have anything other than 1 health, you can't be killed in one hit. Keep this in mind for damage boosting.
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Most hazards maintain their collision even during your invuln frames. Damage boosting through environmental hazards is very rarely possible. Some enemies also maintain collision, but it varies.
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Grabbing a ledge is slow! Land on your feet whenever possible.
Individual Level Notes:
Goon City - nothing to note here. Keeping your momentum from the waterslide at the start can save a few seconds.
Ice Prison - see above about rapid horizontal rope movement. All the snowballs are on global timers, it's very easy to see when you've gone too slowly. There are two very obvious enemy bounce skips to save ample time: one on a bird to catch a vertical chain, another immediately after to go on top of the fire powerup part.
Zorrscha's Lab - Half of the level is easily skipped. After the first cave portion, use the springboard to hop up to the nearby orb. Cross to the right and use the enemy to bounce up to the orb slightly up and to its left. It feels like there should be a way to do something similar to skip the first cave portion, but I haven't managed it yet.
Hot Pants - The only boss with RNG. Takes 5 hits to kill. Only vulnerable when it's in the background; soon after it turns to attack, it becomes invulnerable. Will randomly do a long strafing run and is invulnerable the whole time; a strafing run is generally a reset.
Stan's the Man - Little to note here. Jellyfish sure can eat your jump, so be careful. Stan takes 15 hits from either bouncing or fireball. Fireball is much more reliable and likely faster. He'll turn back after every two hits.
Oyster Desoyster - One of the two levels that starts you off going left, so don't forget. For the helicopter sections, you can purposely run into the underside of a hazard or jellyfish to save time and drop down early, but the positioning is tight. Don't have a reliable setup for them yet.
Puzzlewood - Long and simple. For the miniboss, she takes three bounces to send to the other platform, and three more to defeat. Using the bouncing platform in the middle, you can go from the left to the right after first three hits to be in the air right before she reappears and waste zero time. All of the puzzle piece hazards keep their collision and can easily suck you in to waste time. Avoid at all costs, even if it means waiting a cycle.
Temple of Nori - starts with a long cutscene for some reason. Not much to note here, no real skips to be had. The second grasshopper needs to be killed, everything else can be ignored.
Egg! Egg! - Zero RNG, does the exact same pattern every time. Takes 15 hits. Can be hit a maximum of four times with tight timing while vulnerable. If you get three hits twice, you'll get a 5-cycle, which wastes 30+ seconds.
Huevos Libertad - Egg cages have a heart in them, if you need one. The bouncy room is rough and my setup to get through fast only works 50% of the time: bottom right bouncer, sends you left, hold jump throughout. Should send you back to the right and then up to grab the ledge. Let me know if you have a better one.
Pipe Haus - Tons of horizontal climbing, but the roofs are too low to gain any time by jumping. Can jump to avoid hazards without needing to wait, but timing is tight. You don't need the switch toward the end of the level to make the platform rise; you can make it with a double jump from the bouncy platform.
Hate Tank - Aptly named level. Just the worst. Intensely laggy on most emulators. Holding fire and mashing fire are the same speed, so just hold. Locks take 5 direct hits to break, and the direct hit radius is smaller than it looks.
Fantabulous - The level wants you to use a spring near the start of the big fan room, but you don't need to. Hit the switch to make the platform drop down, and then just double jump over to it. Faster. Cannot skip any switches on the upper portion, even if it looks like you should be able to.
Mr Schneobelen - Pattern is the same every time. Easy to memorize and then guide your shots ahead of him.
Collide-o-Scope - Again, just the worst. You can shield when walking, but there's no reason to. Haven't figured out the exact mechanics of the net robots. Shooting and bouncing on them directly seems to kill them faster, but I'm not 100% on it yet. The ending has an OOB clip you can do to skip the end cutscene, but don't have a reliable setup. You need to bounce on one of the enemies as it charges upward toward the roof, and then be hit by a shot before you hit the ground or bounce on an enemy again. Should clip out into the end of level trigger and skip 30 seconds of cutscene.
The Zoul Train - The only time a lightning powerup is useful. Wizards lock doors and take 5 bounces to kill. Lightning kills them much, much faster, but you need to keep it through very rough platforming to make proper use of it. It also works on Zorrscha and saves immense time in that fight. Keeping the lightning is a difference of over a minute to not having it in total.
Lick the Toad - Nothing much to say here. The race is trivial with proper platforming.
The Bitter End - Memorizing the bouncer locations, the gaps in the main shaft on the bottom half, and when you can leap off of a rail when going up will make all the difference. You can save time with the yellow bouncers in the top half by nudging against the side as you go up and doing a double jump to eat your momentum and land on the next one faster. Need to go up, down, middle, down, and up, all in total. The eye miniboss is easily dealt with by bouncing on the closest platform and timing your hits so that you'll be over its counterattack. Can't double jump with the big boots on, don't try.
Rub the Buddha - All crystals except the big one take 3 hits each. The mouth crystal in the last phase takes 15, and can be hit twice per charge (before and after) with tight aim and timing.