During the film, the main character is cycling through the English countryside listening to his little radio. On the radio is 'The Wanderer' by Dion. The film is set in 1959, but 'The Wanderer' was a UK hit in 1962.
In the last few minutes of the movie, when Rocky comes back to the farm on his tricycle, he is heard yelling "Ginger," but if you took a good look at his beak when you hear him say it, you will find that the beak is not opened. It's closed.
Mac tells Rocky that "You sprained your anterior tendon connecting your radius to your humerus." However, tendons connect muscles to bones and the radius and humerus are both bones, so she should've said 'ligament', not 'tendon'.
In the overhead shot of the huts when Rocky starts teaching the chickens how to fly, hut 24 is behind hut 23. Later, when Nick and Fetcher are watching them jumping on the hot water bottle, huts 23 and 24 are next to each other.
Rocky abseils on a telephone wire through a round window into the shed with the pie machine. The bracket holding the wire is clearly attached exactly above the window. Later, when Mrs. Tweedy involuntarily skydives from the plane, she enters the shed through the same round window (the bracket is clearly visible above it) but she does not hit the wire on her way down.
When Ginger says, in response to Rocky asking who would want to know if he was the rooster on the poster, "A rather desperate group of chickens. You see, if you are...", she starts saying it when the camera is on her. When the shot cuts to Rocky and Ginger's back is towards the camera, Ginger is still heard talking. However, Ginger's mouth isn't moving (you can see that her cheek isn't moving while the back of her head is still in camera).
Puppets were often mounted to rigs so they could be suspended in the air. The rigs were edited out of each frame after filming, but there is a short cut in Chicken Run where a rig was missed:
Inside the pie machine when Rocky is falling out of the meat chute, the 17-frame cut of him falling directly into the camera still has the puppet rig visible.