Visible crew/equipment: In the scene when Nina arrives to her home after she knew she was chosen, she is looking for her mom (who bought a cake for Nina), and there is a camera man in a mirror.
Factual error: Toward the end after Nina dances the black swan and she's in her dressing room, Lily knocks on the door to congratulate Nina on her performance. When Lily turns around to leave, her huge double-wing tattoo is visible on her back. No ballet company will allow a dancer to perform on stage with tattoos that have not been properly covered up.
Continuity mistake: We see Nina put on red lipstick on the subway. In the next scene, when she's in the studio, her lips are skin-colored, implying that the lipstick is off. But when she enters the director's office, the lipstick is on again, even though she never reapplied it.
Continuity mistake: When Nina watches Lily dancing after filling up her water bottle, look very closely at Lily's back as she spins around and you can see that her tattoo isn't there. As she was wearing a vest and her hair was down over her back, it is likely the film makers thought they could get away with this.