Troy drives a Rover 45 with registration plate BX03 ZZG. When Barnaby and Troy learn that Cully may be in mortal danger, they jump into Troy's car and leave the station with a blue rooftop light and siren. But the plate on the car has changed to BU03 ZZG. As they are seen driving with the flashing light and siren, and arriving on-scene, the plate has returned to the original BX03 ZZG.
The goof item below may give away important plot points.
In the very beginning of the episode, we find out that Troy has received his promotion. Even Barnaby congratulates him and calls him Inspector Troy. However, when Troy leaves the note for the homeless man, he signs it 'Sergeant Troy'.