Aside from the usual recommendations of Final Fight, Double Dragon, Kunio-kun, Streets of Rage, Golden Axe, X-Men, TMNT, Power Rangers, and Battletoads, there are some other beat-em-ups that you should definitely check out (Warning: It's a long list):
Big Fight: Big Trouble in the Atlantic Ocean (Arcade)
Zero Team 2000 (Arcade)
Denjin Makai (Arcade and SNES) and its sequel Guardians (Arcade)
Legionnaire (Arcade)
Undercover Cops (Arcade and SNES)
64th Street: A Detective Story (Arcade)
The Rushing Beat Trilogy (SNES; avoid the butchered localization versions)
Sonic Blast Man (SNES) and its sequel Sonic Blast Man II (SNES)
The Combatribes (Arcade and SNES)
Captain Commando (Arcade and SNES)
Battle Circuit (Arcade)
Knights of the Round (Arcade and SNES)
The King of Dragons (Arcade and SNES)
Cadillacs and Dinosaurs (Arcade)
The Dungeons and Dragons Mystara Duology (Arcade and Saturn)
Warriors of Fate (Arcade)
Armored Warriors (Arcade)
The Simpsons Arcade Game (Arcade)
Crime Fighters (Arcade)
Vendetta (Arcade)
Violent Storm (Arcade)
Metamorphic Force (Arcade)
Gaiapolis (Arcade)
Burning Fight (Arcade)
Mutation Nation (Arcade)
Robo Army (Arcade)
The Sengoku Trilogy (Arcade)
King of the Monsters 2 (Arcade and SNES; the Genesis version is a standard fighting game)
Arabian Magic (Arcade)
Dungeon Magic (Arcade)
Growl (Arcade and Genesis)
Warrior Blade: Rastan Saga Episode III (Arcade)
Silent Dragon (Arcade)
Arabian Fight (Arcade)
Spider-Man: The Arcade Video Game (Arcade)
D.D. Crew (Arcade)
Die Hard Arcade (Arcade and Saturn) and its sequel Dynamite Cop (Arcade and Dreamcast)
Zombie Revenge (Arcade and Dreamcast)
DJ Boy (Arcade and Genesis) and its sequel B. Rap Boys (Arcade)
Bad Dudes (Arcade and NES)
Two Crude Dudes (Arcade and Genesis)
Night Slashers (Arcade)
Captain America and the Avengers (Arcade and Genesis; avoid the horrible SNES port)