Note to self, never use a song in which you get an irresistible urge to dance. You will fail miserably at the game and the dance at the same time, no matter how hilarious it is to try.
If you completed story mode before the badges were made, you just need to click on the completed save file and you'll get the badge. At least that's what happened to me.
 "Idle Machine Learning"){kind=icon}
 "Socrates Jones: Pro Philosopher"){kind=icon}
 "Vector Stunt"){kind=icon}
 "Learn to Fly 2"){kind=icon}