How to make animated 2D sprites?

[jackal27]

I have recently gotten into minor stage modding such as adding platforms here and there, changing collisions, moving models around, etc. However, I'd like to take things a step further and start working on some 2D stages. I have a few ideas in mind such as a Kid Icarus NES (and/or Game Boy) stage, an Excitebike stage, and maybe even a Super Mario Land 2 stage, but I've never liked stages that were just static with not much happening visually.

I've seen several stages on Brawl Vault that have animated 2D sprites on them and that is something I would really like to learn how to do. However, I can't seem to find any resources on how to do this anywhere. What would be the best/easiest way to achieve this or to learn how to do it? here are a few stages with animated 2D sprites:

Mario Bros. X
Donut Plains
Outer Wall - Cave Story
Hidden Palace Zone
Retro Brinstar

Thanks for the help!