This took the longest to process, as it's the longest of my 4 Special Stages compilations, so I uploaded it last.
Sonic the Hedgehog 3, Sonic & Knuckles, and Sonic 3 & Knuckles bring back the big Special Stage Rings that were in Sonic 1, and Sonic CD. However, they are much thicker, and instead of finding them at the end of most Acts, they are instead stashed away in hidden areas scattered all over every Zone in the game between Angel Island to Lava Reef. They stop appearing in every Zone after that. If you can find them, you can enter them anytime you want without needing a certain number of Rings.
All of the Special Stages in this game take place on very small checkered planets. The goal is to, well... "Get Blue Spheres". Nab them all to beat the Special Stage and get a Chaos Emerald for your efforts. Once you collect a Blue Sphere, it'll become a Red Sphere. Don't touch those or you're automatically kicked out of the Special Stage.
There's a trick that, if you create a perimeter of Red Spheres in a group of Blue Spheres of 3x3 or more, you'll turn the entire group into Rings. If you collect all of the Rings in the Special Stage, you'll get a Perfect, which gives you 50,000 points: An instant 1-UP!
As with Sonic 2, if you get all of the Chaos Emeralds, you can become Super Sonic and see the game's Good Ending. If you got them all as Knuckles, you can then become Super Knuckles. HOWEVER, since this is Sonic 3 & Knuckles, you can go further than that.
Launch Base is the last "Sonic 3" Zone, and in this game, it leads directly to Mushroom Hill Zone, the first "Sonic & Knuckles" Zone.
The big Rings you'll find in those Zones flash different colors. Enter one and you'll be taken to Hidden Palace Zone's Emerald Shrine. From there, any number of Chaos Emeralds you had at that point will be taken from you and transformed into unactivated Super Emeralds. This means you're no longer able to become Super Sonic/Super Knuckles. So if you still want to beat the game as just them, you're going to have to avoid every big Ring from there on out!
To reactivate the Super Emeralds, stand on top of a dull grey one to be taken to its Special Stage. The Super Emeralds' Special Stages are the exact same ones as the Chaos Emeralds' if you're playing Sonic & Knuckles on its own.
A new addition in these Special Stages are Yellow Spheres, which act like springs that give you a big jump that skips over several spots in front of you.
The color order for both Chaos & Super Emeralds are: Green, Orange, Pink, Blue, Grey, Red, Cyan.
Winning a Super Emerald causes its essence to find itself at the shrine, reactivating the emerald with its matching color.
If you get all of the Super Emeralds, Sonic can become Hyper Sonic, Knuckles can become Hyper Knuckles, and Tails finally gains the ability to become Super Tails. Sonic and Knuckles' Hyper forms are more powerful than their Super forms, but Super Tails has a very unique quirk about him that probably outperforms the two Hypers, especially against bosses.
Also, you get to see an even better ending once you beat the game. A Perfect Clear of sorts.
Last but not least, there are two extra Emeralds, as well: A Yellow Chaos Emerald, and a Yellow Super Emerald. To reach their Special Stages, you have to first unlock the Stage Select & Debug Mode, then go to the former's Sound Test and go to "1F" on it.
Then Hold A and press Start after highlighting "Special Stage" on the same screen, "Special Stage 1" for the Yellow Chaos Emerald, & "Special Stage 2" for the "Yellow Super Emerald.
Besides being able to carry over the stats of the last data you picked (such as having all 14 Emeralds), both of these Yellow Emeralds don't do anything in the game besides adding an unnecessary Emerald to your total, and once you get them, you're automatically taken to the beginning of Angel Island Zone.
I'll cover how you can unlock Stage Select & Debug Mode in the first place in a different video, as well as how you can do the same in Sonic 1, 2, & even CD!
9/8/2017 UPDATE: Hey! It looks like ALL 16 of these Special Stages make a return as Sonic Mania's Bonus Stages! These 16, combined with 16 brand new layouts, make a grand total of 32 regular layouts! I will record all 32 of them eventually!
Sonic 1's Special Stages:
Sonic 2's Special Stages:
Sonic CD's Special Stages (+1):
Sonic Mania's Special Stages:
Sonic Mania Plus' Encore Mode Special Stages: