Continuity mistake: Amy shows up at Oscar's place to ask if she can move in. As his friend's leave, watch closely and you can see that they exit twice. While there is also a hallway opening, you can see that the black guy, Bill, is already halfway through on the first shot. The other two are spaced behind him. The scene cuts away and all three guys are just exiting the apartment door and are close behind each other.
Continuity mistake: When Oscar is almost caught listening to Amy singing in the shower, he tries to get away from the door quickly and in doing so trips over the hat rack, knocking a scarf off onto the floor. The next time we see the hat rack and its surrounding area, the scarf is nowhere to be seen.
Continuity mistake: When its pouring rain and they get in the second taxi, after the first one breaks down, that taxi is a newer styled yellow cab. In the next shot, the cab is an older style, box yellow cab, then it switches again to the newer style, in the next shot.