Most people (or at least most that I've talked) to say that their favorite beat 'em up is Streets of Rage 2. And that's an excellent choice. The game is fantastic. But for me, I'd have to give the nod to Capcom's 1989 arcade hit, Final Fight.
My first time playing Final Fight was on the SNES. At the time, I didn't know that it was a port of the arcade game and that there were quiet a few things missing, like an entire level, the transexual Poison, co-op support, and of course, my main man, Guy. Even so, I dug the game. I didn't play the arcade version until 2005 when Capcom released their Capcom Classics Collection vol. 1 compilation. My love for Final Fight grew tenfold when I played this game's version.
If you haven't played it, Final Fight really isn't different from any other beat 'em up, because let's face it, the genre as a whole, really isn't all the deep to begin with. But it satisfies the need to walk to the right side of the screen and kick the crap out of thugs with a piple, knife, or a katana. Mindless fun at it's best. What makes Final Fight such an awesome game is the execution. None of the stages are too long or too short, Code, Hagar and Guy are all fun characters to play as and since the game was made by Capcom, one of the best fight crafters in the biz, you know you've got a hit on your hands.
My favorite version of this game is naturally the arcade version, but that doesn't mean I wouldn't recommend other version. The SNES version has some nice music, some would say even more fleshed out than it's arcade version and the Sega CD version is one of the best out there and has an incredible arrange soundtrack. There's also Final Fight One on the GBA, which has all the games levels, all three playable characters and some extras like Code and Guy's Alpha clothing. Even better, when you play using Alpha Cody or Alpha Guy, both characters acknowledge the fact that they are back in the past.
So yeah, Final Fight rocks. It's my all-time favorite beat 'em up and one of my all-time favorite games. You haven't live until you've played this one.