Continuity: When Jean is looking at Charles in the mirror, what she sees is the right way round (you can clearly see this by looking at the cover of Charles' book).
Errors made by characters (possibly deliberate errors by the filmmakers): When Eve is presented by her Uncle at the party, she is referred to as "Lady Sidwich", but her actual title is "the Lady Eve Sidwich".
Revealing mistakes: When Charles and Jean are standing on the deck talking, the ambient, background sounds - of people talking and birds singing - can be heard running on a loop. The scene lasts several minutes, but the ambient sounds repeat every several seconds.
Crew or equipment visible: When "Colonel" Harrington talks to Jean in her cabin the morning after Pike proposes, the shadow of a spotlight moves up and out of frame.