Revealing mistake: Cary Grant shows off his ice skating moves but you can tell it's a stunt double. The double is a bit shorter than Grant. Also in the same sequence, the two other skaters have noticeable doubles as well.
Character mistake: In the final sermon, Henry says "Uncle Henry" instead of Uncle Harry - as shown in the earlier typed sermon.
Continuity mistake: When Dudley drafts the sermon he says "you give me a tie, I give you a book." When the bishop reads the sermon he says "you give me a book, I give you a tie."