Slingshots are the most commonly used glitch in Any%.
They combine magdashing with HSP, so it's often recommended to learn these first, though unlike HSP, slingshots aren't frame perfect.
To perform a slingshot, you must start by magdashing on an enemy, and targeting at any time during that. This will allow you to get further away from the enemy than usual while still targeting it. This is usually done in the order: Homing attack, Boost, Target.
After that, land on a floor, then perform the inputs for an HSP: Dodge, Stomp, Boost.
The timing to press the boost button after stomping is from frames 7 to 14 after starting the stomp attack.
Frame 7 specifically is referred to as a "frame perfect slingshot", which is usually undesirable, since it gives you a lot of height and that's often slow.
When slingshotting on the spinning enemies, the timing starts on frame 4 instead of frame 7.
When doing it on 30fps, the timing windows are calculated on 30fps frames, and as such, you will have to wait double the amount of time from 60fps.
This tutorial by Onaku explains slingshots in depth, though it's worth noting that all of the routing examples given in the video are outdated:
A common issue when learning how to slingshot is that you will accidentally stop targeting the enemy, known as "breaking the target". This needs to be avoided, and can happen due to various reasons:
If a platform gets between Sonic and the enemy, the target will break. This can include floating platforms, trees, or even part of the island's terrain if you're on a hilly area.
You need to ensure that nothing gets in the line between Sonic and the enemy.
If you go too far away, the enemy will despawn. Make sure you get enough practice on each slingshot to know how far you can safely run before that happens.
If you take too long between boosting and dodging, the target will break. When starting out, it can help to avoid releasing your boost until you're dodging.
If you move the camera too much in a busy area, it can cause you to accidentally start targeting something else.
You can do slingshots off of the bird enemies in Chaos Island, but instead of magdashing, you need to stomp them and then boost away before they die.
Note that you can only stomp birds before you aggro them, and you can do so from a really long distance away, as the game will move you over so that you are on top of the bird.
Also keep in mind that the uneven terrain in Chaos Island can easily cause the target to break.
When you need to get more height and distance off of soldiers, you can start a slingshot by performing a cyloop around the enemy and then boosting away. The cyloop will launch the enemy into the air, which will change your slingshot's speed.
If an enemy is attacking while you cyloop it, it won't get launched into the air, so it's best to do the cyloop while the enemies are spawning in.
It's easy to accidentally target a different soldier from the group if you target at an arbitrary time, but targeting right after releasing the cyloop ensures that you will target the correct one.
In some situations where you'd want to do a slingshot, there's no easy way to land on the ground after magdashing off of an enemy. This is mainly seen in Rhea Island, where all the ground you have to work with are the small platforms on the way up the towers.
Stomping after the magdash will allow you to land on a platform, though before doing this, you should familiarize yourself with the game's depth perception.
For this type of slingshots, you should make sure that you hold neutral while you're touching the ground, otherwise the target will break.