The Parent TrapGoofs
When Elizabeth is packing for her trip to meet Nick and switch back the two girls, Hallie comes into Elizabeth's room and tells her that she just talked to Nick and that he's excited to see her. After Elizabeth starts to freak out about the fact that Nick is "excited" to see her, she tells Hallie to go ask/see her "grandmafather." You can clearly tell that she says that and you can tell she realizes she said it because she turns around really quickly.
Crew or equipment visible
Errors in geography
The opening sequence set aboard the ocean liner QE2 was obviously filmed aboard the museum/hotel ship RMS Queen Mary in Long Beach, CA. The equipment on the bridge shown in the movie (brass steering wheel, etc) is part of the Queen Mary tour and is far to antiquated to be in service on the QE2 in 1986. The dark wood interiors are a giveaway as well. Finally, when Nick and Elizabeth are looking up at the fireworks, you can see the coin operated binoculars from the Queen Mary in the background along the rail. The QE2 did not have those on her deck.
When Hallie wins at poker the first time the twins play, she produces a royal flush and and says "in your honor" to Annie, obviously referring to her British nationality. However, the music being played in the background is the first notes to La Marselleise, which is not only the French national anthem, but also revolutionary and anti-monarchic.
Incorrectly regarded as goofs
Meredith tells Annie (as Hallie) that she is going to marry Nick in two weeks and nothing "Hallie" does can stop it. Later in the movie, they are supposedly at the Stafford and trying to decide if they want to have the wedding there. If the wedding was less than two weeks away, the invitations would have already been sent out with a location on them.
When Meredith meets Elizabeth at the bar, she glances at Elizabeth's bill and says, "You're Elizabeth James?" and then begins talking about how she's designing her wedding dress. Elizabeth James is a common enough name that Meredith would not necessarily have known she was talking to the same Elizabeth James.
When Nick and the girls return from the camping trip, they have a brief conversation with Liz and Nick before heading inside. During this conversation, when Nick is viewed on the left hand side of the screen, he is reflected in the glass of the door they are standing in front of. Although one of the twins is standing right next to Nick, we do not see her reflection, only Nick's. Even when she passes in front of the door, her reflection doesn't appear. (This is most likely a result of the camera tricks used to make Lindsay Lohan appear as twins.)
When the girls are in Meredith's tent during the camping trip, their figures are only silhouettes. One of the girls says something, and the other replies but the sound comes from the same source, proving that in that scene the crew used a lookalike (probably a stunt double) and had Lindsey Lohan say both lines.
Early in the film, Annie and Hallie begin to discover their similarities, one of which is that they both like Oreo cookies with peanut butter. However, Oreos were not widely available in the UK in 1998; they started to become available in packs of four about five years after this film was made. Growing up in England, Annie would probably never have tasted an Oreo.