Joystiq has posted the music list used in BioShock. Obviously these are standards used in the game, not the score masterfully crafted by Garry Schyman.

From the article:
’20th Century Blues — Noel Coward
Academy Award — Stanley Black
Avalon — Django Reinhardt
Bei Mir Bist Du Schon (means you’re grand) — The Andrews Sisters
Best Things in Life Are Free — The Inkspots
Brother Can You spare a Dime — Bing Crosby
Danny Boy — Mario Lanza
God Bless the Child — Billie Holiday
How Much is that Doggie — Pattie Page
If I didn’t care — The Ink Spots
It Had to Be You — Django Reinhardt
It’s Bad For Me — Rosemary Clooney
Jitterbug Waltz — Django Reinhardt
Just One Of Those Things — Lee Morgan
Just Walkin’ in the Rain — Johnny Ray
La Mer (Beyond the Sea ) — Bobby Darin
La Mer (Beyond the Sea) — Django Reinhardt
Let’s Fly Away — Lee Wiley
Liza — Django Reinhardt
Night and Day — Billie Holiday
Papa Loves Mambo — Perry Como
Please Be Kind — Django Reinhardt
The Party’s Over Now — Noel Coward
The Waltz of the Flowers — Pytor Llyich Tchaikovsky
This Is A Changing World — Noel Coward
Wild Ride — Faux Frenchmen
World Weary — Noel Coward
Wrap your troubles in dreams — Bing Crosby
You’re Getting To Be A Habit — Harry Edison Bing Crosby
You’re the Top — Cole Porte’