Last Blade was the first game to come to mind. I couldn't think of any specific stages, I just remembered it having awesome backgrounds. Also SF3: 2nd Impact had some amazing backgrounds. I don't know why they didn't include them in Third Strike.
The SFA2 stage that takes place in a bathroom... can't find any good pics online.
Also was looking through the SNK wikia, lots of fantastic sprite-art backgrounds. I like this one because, for some reason, random Japanese games would have "NEW JACK" in their grafitti (ESP Ra.De. has this too)
1. Capcom hires a company to do all the in-between animation cells for the game.
2. Sprites are pretty dang awesome, tons of shading, very detailed.
3. The animation company cannot keep up with the detail, and is very inconsistant.
4. Capcom/Udon start lowering the ammount of detail in hopes of getting it done faster.
5. Animation company still sucks.
6. Capcom fires them, then has to re-do all the animation over again, with a new company.
8. They have the reduce all the detail/shading so they can get it done relatively on-time.