Continuity mistake: When Bugs is calling the operator on the telephone, you can see there is a picture on the wall. The picture disappears in the next shot, and in the shot after that, it is back again.
Continuity mistake: When Elmer says "Step right this way" before escorting Bugs out of his house, you can see the handle is on the right hand side of the door, but when we see Bugs in the next shot on the other side, the handle is on the right hand side as well. This means that the door handle where Elmer is should have been on the left.
Continuity mistake: When Elmer enters his house, you can see the front door has one long panel on it. But when Elmer manages to get Bugs out of the house, the door now has two small panels on it.
Continuity mistake: When Elmer opens the door and finds Bugs at the dressing table in ladies' underwear, you can see the door is positioned on the left hand side of the frame. But when we see Bugs, the door is now on the right hand side.