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Don't know many of either consoles sadly, but as for the ones I do, the ones I'd recommend:
There are a lot of good ones. I'll just highlight Streets of Rage 2. Plays great, music is great, looks good. It's my favorite game on either system.
X-Men 2 on Genesis
The list would go on and on. So to keep it concise:
Action & Platformers
Castlevania: Bloodlines
Contra Hard Corps
Kid Chameleon
Shinobi III: Return of the Ninja
Sunset Riders
Vectorman 1 & 2
Beat 'Em Ups
Alien Storm
Golden Axe Trilogy
Streets of Rage Trilogy
Role-Playing
Beyond Oasis
Crusader of Centy
Phantasy Star series
Shadowrun
Sports & Racing
NBA Jam
OutRun
Road Rash
I could keep naming games, but that should be enough to give you a taste of what made the Genesis unique, as well as some spins on franchises that dominated the NES.
Oh, check out Toejam and Earl as well.
Master System: Phantasy Star
Genesis: Phantasy Star 1-4
Gain Ground
Altered Beast
Golden Axe
Revenge of Shinobi
Starflight
Eternal Champions
Sonic 1-3 + Knuckles
Comic Zone
Streets of Rage 1-3
(Bare Knuckle 3 for the uncensored version)
Sega CD: Snatcher
Eternal Champions (re-worked from the cart version)
Sonic CD
I'll suggest some great games on master system, I could go on all day about Sega.
Sonic. The 8 bit version is not a port of the 16 but game, it's a master system exclusive and I've always liked it a bit more.
Wonderboy in Monster Land. It's true Wonderboy 3 is a better game, but it also has a stellar modern remake. Monster Land for me is the Wonderboy go to if I'm during up the SMS. And Monster World for mega drive of course.
Asterix. Just my favourite game when I was a small kid. Pretty much just perfect platforming and if you loved Asterix comics it's really cool to see it as video games.
Golvellius. An action-adventure, I guess you can liken it to early Zelda games with top down perspective and since side scrolling. Pretty hard but good fun.
Lucky Dime Caper. Another platformer, but Sega killed at making these perfect. It's Donald Duck.
The Ninja. Top down, like, shooter, I don't know what the genre is called. Like Ikari Warriors/Commando/etc. but actual fun and really good. Needs strategy, quick reflexes and hunting secrets.
Fantasy Zone 2. My favourite shmup on the system, better than the first one for me. It's a cute em up. There's also Fantasy Zone The Maze which is a somewhat Pacman like game set in this world. I played that a lot too, definitely fun and good in 2-player.
Super Monaco GP and World Grand Prix are the racing games I like. 8 bit racing games are simple, but they're fun.
Tazmania. Another great platformer, this one has Taz from looney tunes. This is streets ahead of the 16 bit version and it's not even close.
Cloud Master. A shmup I used to love. Going back now I can barely beat the first stage and it has some slowdown so YMMV.
Double Dragon. The beat em up for master system. Looks great. Plays, fine, hit detection is a bit wonky but it's not bad. Great translation of the arcade game.
Robocop vs Terminator. Great on both Sega systems.
Aztec Adventure. Like Golvellius, it's a bit early Zelda like, much more straight forward though. You can bribe some enemies to help you which is interesting mechanic.
Rock 'n' roll racing on the Genesis still holds up as the finest co-op combat racer ever made, IMHO.
Starflight is neat if you like space exploration.
Slime World is a fun co-op side scrolling shooter.
Windjammers is a fun disc-soccer game, kinda like extreme pong.
I'm not familiar with the master system, but the Genesis has a fair number of good games. I think it unfortunately doesn't hold up as well as it's competitor the SNES though, because Sega went hard on licensed properties and sports titles, and a lot of those games just aren't particularly relevant these days, and there were a lot of cross platform games that were just better on the SNES.
But a few that I think are worth mentioning:
That's a great list! I disagree with SNES holding up better though. For RPGs yes, but anything else you better go Sega.
I want to suggest some more great Mega drive games:
Haunting starring Polterguy
General Chaos
Mutant League Football
Subterrania
Desert Strike, Jungle Strike
Ranger X
Flink
Shining Force 1 and 2
Flashback
Skeleton Krew
Shadowrun
Wonderboy n Monster World
Skitchin
Phantasy Star 4
All the Ecco games, even Ecco Jr which is just very relaxing to sit and play through
This is an awesome list and I just want to add a couple more. The SNES and Genesis each had a Shadowrun game but they're totally unrelated. I think the Genesis version is better. Same with Pirates of Dark Water (I loved this show). The SNES and Genesis versions are completely different and I prefer the Genesis game.
Other than that, I really like Elemental Master. It's a vertical shoot 'em up but you play as a mage, always walking up. So it's kinda like the NES game Gun.Smoke if that means anything to you.
Also have a look at Battle Mania: Daiginjo for Genesis / Mega Drive. A side scrolling shmup where you control anime girls. It's Japanese, but you actually don't need to understand. Otherwise there should be translations available.
I can't believe it hasn't been mentioned yet. Get Ristar. Awesome game! And also Soleil (in the US, it had a different name but searching Soleil should give you the US name).
Crusader of Centy in the US.
My favorite Master System:
Phantasy Star
Zillion
Impossible Mission (Commodore 64 is a better port though)
Honorable mentions: Golden Axe Warrior (Zelda clone), Vampire: Master of Darkness (Castlevania clone).
Uh, for Genesis my list would be too big lol
I absolutely love Shadownrun (completely different game from the SNES version, both are awesome but I prefer this one)
If you like Alien³ you will most probably like Jurassic Park and Jurassic Park Rampage
Kid Chameleon, Landstalker and Beyond Oasis are pretty good exclusives
Phantasy Star and Shining Force series are great RPGs. Streets of Rage 2 is an awesome Beat'em Up, and so is the whole series. Golden Axe series also quite fun. Teenage Mutant Ninja Turtles: The Hyperstone Heist is a shorter and tougher version of SNES' Turtles in Time. Contra Hard Corps is also pretty tough and very fun. Castlevania Bloodlines too. Some other worth checking Gunstar Heroes, Alisia Dragon, Mega Turrican, The Killing Game Show, Vector Man 1 & 2, Comix Zone.
In general, even though I hate the characters, Disney games are great. Mickey's Castle of Illusion and Land of Illusion for Master System, the Castle of Illusion version for Genesis and also World of Illusion, and Quackshot, are all quite good.
Some of my sms faves: Outrun, Space Harrier, Phantasy Star, Rampage, Psycho Fox, Spellcaster
You will get lots of games recommended, so I will make it short and only mention something that is not talked already in this thread.
Mega Drive / Genesis
Master System
SMS games I enjoyed as a kid. Double Dragon, Shinobi, Rambo, Outrun, and Choplifter.
As a place to start, here is what I think is a well curated list of top titles for many retro systems including the SMS and Genesis.
My favs are Ecco that can be a bit frustrating sometimes but is fun overall, earthworm Jim a goofy side scroller, eternal champions and mortal Kombat for fighting games, NBA jam arcade baller, 90s nhl loop around the back of the net, mutant league football and hockey, streets of rage brawler with amazing soundtrack, and you already mentioned the sonics. Phantasy star 4 for an rpg maybe not the whole game I think it got pretty grindy but at least the first couple missions.