Here is the full list of all 36 Sonic Origins achievements.
Unlock all Trophies
Play Mirror Mode
Defeat 10 Moto Bugs in Sonic the Hedgehog
Defeat 10 Rhinobots in Sonic the Hedgehog 3 & Knuckles
Clear 10 Sonic the Hedgehog 3 & Knuckles Missions with an S rank
Clear Sonic the Hedgehog 3 & Knuckles in either Anniversary or Classic Mode
Clear all 4 titles
Try the Boss Rush from any of the titles
Unlock 10 sound collection items with Coins
Unlock 30 art collection items with Coins
Unlock 6 movie collection items with Coins
Clear 10 Sonic the Hedgehog 2 Missions with an S rank
Clear 10 Sonic CD Missions with an S rank
Clear 10 Sonic the Hedgehog Missions with an S rank
Fly as Tails
Clear Sonic the Hedgehog 2 in either Anniversary or Classic Mode
Get the jackpot in Sonic the Hedgehog 2's Casino Night Zone
Turn into Super Sonic
Defeat 10 Stingers in Sonic the Hedgehog 2
Watch the opening for Sonic the Hedgehog 2
Clear Sonic CD in either Anniversary or Classic Mode
Win a race against Metal Sonic in Sonic CD
Clear Sonic the Hedgehog in either Anniversary or Classic Mode
Defeat a total of 200 enemies
Use Air Bubbles in Sonic the Hedgehog
Collect a total of 1000 Rings
View a Premium Collection item from the Museum
Collect 3 types of Shields in Sonic the Hedgehog 3 & Knuckles
Watch the opening for Sonic the Hedgehog 3 & Knuckles
Defeat 30 enemies with the Spin Dash
Defeat a total of 50 enemies
Clear 1 Mission
Travel to the past or future in Sonic CD
Watch the opening for Sonic CD
Glide as Knuckles
Watch the opening for Sonic the Hedgehog