The temporary fix for the infinite loading screen issue
Okay, since people are yapping about how Rockstar didn't fix this issue, this topic is being temporarily pinned until Rockstar sorts this problem out.
There was a fix for this since day one, and it's really simple;
What you have to do is disconnect from the Internet, or even better, sign out from the PlayStation Network/XBOX LIVE.
You can use BAWSAQ (witch only works if you're signed in), by singing in while the game is loaded, you can save your game and do anything you wish, you just cannot load another savefile until you restart the game.