As far as US releases are concerned,
On the NES
Pro Wrestling on NES still holds up extremely well. Exceptionally well IMO. It's a shame that some of the characters/actions have kept Nintendo from re-releasing it on Virtual Console or elsewhere.
Of course Pro Wrestling owes practically everything to Taito's arcade classic Mat Mania, which also holds up extremely well and similarly has characters (Coco Savage) that just wouldn't fly in an era of appropriate racial sensitivity.
Tecmo Wrestling is good, some impressive presentation (cut-away cinematics on power moves) and power-up training system.
The WWF games on NES are all horrible. With licensed property king LJN at the helm, play mechanics are all but an after-thought.
WCW Championship Wrestling seems to have the seeds of good ideas (large roster, actual wrestling play mechanics) but it's just so stiff and ugly that it's not worth the time/effort required to learn how to play it well.
M.U.S.C.L.E. gets a bad rap, it's not a technical marvel by any stretch of the imagination, but as a two player "fighting" game, it's certainly possible to have fun with. If nothing else it created a video game legacy that got us to the Ultimate M.U.S.C.L.E. games on GameCube and PS2 which are pretty fantastic IMO.
Tag Team Wrestling is the bottom of the barrel for NES wrestling games, however - according to the Chapman Bros. who created him, the HomestarRunner character StrongBad got his name from the masked "Strong Bads" from that game. I do love me some StrongBad so I'll tolerate Tag Team Wrestling's existence for that.
*Same goes for Tag Team Wrestling on Sega Master System with the exception that it's a tiny bit better looking than the NES version.
On the SNES:
The WWF games improve to some degree with the peak of quality being Royal Rumble. Good graphics, decent play control, finishing moves, etc.
WCW SuperBrawl Not sure if this game uses early Fire Pro Wrestling tech, but it suffers from similar problems to its NES junior. It LOOKS a lot better, but it's generally too stiff and requires a lot of learning to get the most out of the wrestling mechanics.
Hammer Lock Wrestling is VERY similar to Tecmo Wrestling on NES. Very nice presentation with close-up large sprite cut away "cinematics". An odd one, can be hard to find in the wild, but worth grabbing if you like wrestling games. Like a lot of unlicensed wrestling games of this era it has aesthetic equivalents of current wrestlers of that time like Lex Luger, The Road Warriors, etc.
Natsume Championship Wrestling More arcadey than other unlicensed wrestlers on SNES, but sadly another one that's tough to play due to some really terrible collision detection and very challenging computer AI.
But, anybody who knows me knows that if there's one wrestling game in the 8/16 bit era that I hold above all others, it's:
Saturday Night Slam Masters Capcom's amazing hybrid wrestling/fighting game. Excellent on both SNES and Genesis though the Genesis version trades 4-player multi-tap mode for a 1-on-1 electric barb-wire death match. With character designs amusingly reminiscent of WWF and WCW wrestlers of that era by Tetsuo Hara (of Fist of the North Star), incredibly balanced/engineered play mechanics that blend the best things from wrestling games of that era with the best things from Street Fighter games, tons of "nods" to Capcom fans (appearances from Street Fighter charachters in the crowd, etc.) and a great soundtrack by Capcom regulars Toshio Kajino and Shun Nishigaki.
For my money Saturday Night Slam Masters, any way you can get it - SNES, Genesis, or on a good MAME/CPS emulator (it's CPS1, so it typically runs beautifully in emulation) is the way to go with 16 bit wrestling.
I was actually a "challenge" competitor at PAX East last year. Anybody who could best me in a game of Saturday Night Slam Masters got a PAX Challenge Coin. Took the better part of the weekend for somebody to step up to the challenge, but instead of getting bummed about losing - I was actually thrilled to find somebody else who loved the game as much as me when it finally happened. Turns out the guy who beat me was a staff writer who did a review of the SNES version for a gaming publication in the late 90s (can't remember if it was GamePro or EGM) and wound up similarly falling in love with the game.