Here is the full list of all 25 SEGA Mega Drive and Genesis Classics achievements.
Columns - Gain a high score of at least 5000 on easy
Complete 1 Challenge
Sonic The Hedgehog - Collect all 6 Chaos Emeralds
Golden Axe 3 - Max out your magic
Relaxin' all cool
Alien Soldier - Use 4 Different Weapon Modes in one Session
Golden Axe - Use Magic When No Enemies are Present
Space Harrier 2 - Reach a Score of 5,000,000
Toejam and Earl - Dance the Hula
Sonic Spinball - Collect all 3 Chaos Emeralds
Golden Axe 2- Kill 25 enemies by throwing/knocking them off the environment in a single play through
Dynamite Headdy - Defeat the Robo Collector
Sonic 2 - Finish Emerald Hills with 5 Lives
Shining Force - Make it to Alterone
Complete 10 Challenges
Shinobi 3 - Get a Shinobi bonus (complete a level without throwing a shuriken)
Streets of Rage - Get to the final boss without using a continue
Alex Kidd - Win 50 Games of Rock, Paper, Scissors
Shadow Dancer - Complete the first two stages without using magic
Streets of Rage 3 - Finish the game on Normal or Hard difficulty
Golden Axe Games - Collect 200 magic
All Challenges Complete
Play 20 Games
Play 50 games