Here are the codes that work for all 4 characters. Each variation has 2 different codes that you can use, but both codes do the exact same thing:

These codes seem to only work with Shovel Knight:

This code changes your color at every room transition.

This code changes your color every time you take damage.

This code changes your color every time you jump.

These codes should all allow earning Feats, and should work with all characters!

How do I input a cheat?
When starting a new file and choosing a name, instead type in one of the cheat codes to activate it on that file. Cheats cannot be stacked, so only one cheat per file. This also means cheats can't be activated on an existing save file, only a new one.

Do armors/cloaks overwrite these colors?
No, you will keep whatever palette you choose no matter what armor/cloak you equip. The very expensive armors/cloaks will still have the sparkles/animations/ect. while keeping your cheat code color. Even in Specter of Torment the Donovan Set will be in the palette that you choose.

Does this work with gender swap?
Yep

Does this work on all consoles?
Yep again,
Although, it should be noted that if you're playing the 3ds version (and probably Wii U version as well), this will count as a "cheat" and won't allow you to earn feats with that playthrough.

Does this work with Amiibo Costumes? (Only on Nintendo Consoles)
Yes and no, every one of these colors can be used with the Amiibo costumes, however both the costumes AND each palette need to be unlocked by leveling up your custom knight. So you can't just scan the Amiibo on your save file and thats it, you have to be using the custom knight and select the costume and palette. This also means you don't really need these cheat codes if you have a decked out custom knight saved to your Amiibo.

If I made any mistake in this guide, just comment to let me know and I'll fix it :D

Have fun and stay colorful!