You can unlock all the Karaoke songs by progressing in the story (dating Tiffany) and doing a favor (called "Real Men Don't Sing Karaoke").
The game is really easy, just move the the left stick up and down along the lines of the song, there's no singing and no real challenge.
The one thing to know is if you replay a song your high score will be updated even if you didn't beat it. Because of this your most recent play of each song needs to be over 90 percent. The best way to do it, is block out about 30 minutes sit down and play every song as fast as possible. They trophy will pop after the final song.
Here is a track list to confirm you got them all.
I Fought the Law - The Clash
All Out of Love - Air Supply
Reelin’ In The Years - Steely Dan
Hit Me With Your Best Shot - Pat Benatar
Bad Case of Lovin’ You - Robert Palmer
I Ran - Flock of Seagulls
Girls Just Wanna Have Fun - Cyndi Lauper
Take On Me - A-Ha