This might be a bit broad [but answerable in a broad sense], and I have no documentary evidence to back things up, but my qualification for this is I am a (retired) professional sound engineer, who now works in the movie industry...
Back in the old "Singin' in the Rain" days, the music would be completed first and the actor would lip-sync to a playback on ...
I believe that the cast did in fact sing all of the songs in the movie. Some are even professional a capella singers.
From People's Choice
Everyone in the films who plays a singing part had to sing a full song
for their audition.
Any time protagonist Beca, played by Anna Kendrick, was singing alone, she was singing live on set.
Kelley Jackle, ...
According to this article from Post Magazine, there were no instruments, but there were guitar, bass, and drum vocalizations, plus a beatbox. With the vocalization technique, apparently you are singing through an instrument, so it is still your voice rather than the instrument that is causing the sound (think Peter Frampton and his talk box).
After listening to the soundtrack for this movie, I firmly believe now that this may have been a directorial mistake. In the soundtrack, the Treble's steal the song at the appropriate time. However, as stated, in the movie, it's about 4 or 5 words too late.
The lyrics for song being sung by the Bellas:
'Cause I may be bad, but I'm perfectly good at it
Sex in the air, I don't care, I love the smell of it
... and by the Trebles:
Sex, baby. Let's talk about you and me.
So it looks kosher to me as the guy riffed off the next word, sex, which was just about to be sung.
If you're talking about the scene that starts around 1:06:30 (on my copy, anyway) when she's running into the campus radio station and the station manager offers her a slot in the lineup, it looks like it's a remix of Bulletproof by LaRoux and Release Me by Agnes:
Firstly, Beca wanted to go to Los Angeles and become a music producer but her father refuses it and forces her to become a student at Barden University in Georgia. If you've observed Beca's character keenly, in the beginning she's a recluse and doesn't want to make any friends. On the other hand, Jesse likes Beca and wants to make her a socially acceptable ...
While technically the recording is true and all based on vocalization, in a competition setting, it wasn't accurate. There weren't enough girls on stage to do the complexity of the song. There were so many layers of sound that's possible in a studio but not in a stage setting with the limited number of girls on stage. You can't be simultaneously singing ...