During the bike tests, when Jeremy uses James' bike to re-take his test, you can clearly see that the statue (Darcey), they bought for James isn't on the back of the bike while Jeremy is riding it, but when he gets off the bike after failing again and lets the bike fall over, she is tie on to the back and breaks Darcey's head off.
After the boys fail to buy cars, and James and Jeremy meet with Richard at the Vietnamese café. Jeremy is carrying his water-sodden shoebox. He puts the box down on the table. However, he goes to pick up a chair, and in the next shot, he goes to sit down, he is carrying the shoebox again.
When the backup transport bike is first revealed, it has two US flags on the back and the helmet is blue with silver stars. On the Hai Van Pass, a crew member is seen riding the bike behind James May (after he shouts "Don't leave me!"). However he is wearing a solid grey helmet and the two flags have disappeared.