Goldmember mentions both The Bad News Bears and The Village People. The scene is supposed to be set in 1975. While he may have seen a coming attractions ad for the former, The Village People did not form as a group until 1977.
When Austin enters Goldmember's club in 1975, Foxxy Cleopatra is singing a song that includes a portion of KC and The Sunshine Band's "Shake, Shake, Shake" which wasn't recorded and released until 1976.
When Austin and Nigel are in the car arguing, seconds before they about to hit another vehicle Foxy tells them to "Shut up and turn around please!" But her mouthing movements do not match with what she is saying.
When welcoming Mini Me following his defection, Number Three welcomes him to the Ministry of Defence however the British secret service is SIS or MI6, not the Ministry of Defence, which is in charge of the military.
While Frau does say Scott just wants to make his father proud, it is unlikely that he would turn evil, considering how rocky their relationship was and how against Scott was of his father's evil empire.
Dr. Evil is explaining his plan to use the tractor beam to attract the meteor to melt the polar ice caps. In the video, you see Midas 22 hitting the northern ice cap and flooding the world. There is no land mass, or "ice cap" on the north pole.
When Dr. Evil turns good at the end, this is highly unlikely. He was raised evil and spent 6 years in evil medical school, so it is very unlikely that he would change so drastically in just a few seconds.
In the first scene where Dr. Evil is revealing his plan for Preparation H, the camera cuts between him and Scotty a few times. Each time, the people standing and sitting on the couch in the background are in different positions.
When the globe lands on Dr Evil in the submarine lair, a seaman is standing on the right, holding a huge spanner (wrench. In the next shot, another seaman is standing beside Dr. Evil, and the spanner has disappeared.
After Goldmember escapes with Nigel Powers to 2002 in Dr. Evil's time machine, Foxxy and Austin run out and Foxxy is wearing sneakers instead of her gold platform shoes. When you see them again, Foxxy is wearing her gold platform shoes.
In the opening action sequence, right when the Shaguar slams on the brakes allowing the helicopter to pass over, the markings on the road switch back and forth between a double yellow line and a single broken line.
When Dr. Evil has just had the globe crash onto his head, he is standing straight at the camera. Then when the shot changes and the moon is about to hit him his footing changes turning him slightly towards the right.
After Dr. Evil has stopped the laser and electrocutes Goldmember in the process, Goldmember falls over the edge to the floor, but his position doesn't match the way he fell. He fell forward and did a slight flip, and should've been laying on his back, face up, but instead is face down, sprawled on his stomach.
In the opening "action" sequence of the movie, when the camera first pans over the "Shaguar" to show it is remotely controlled, you can see a reflection of a camera on the end of a white crane filming alongside the car as it drives along.
During the opening sequence, when the camera pans over the remote-controlled Shaguar, the driver's seat padding is noticeably bulging from the hidden stunt driver, and the driver's left arm can be seen down by the door.
When Austin goes to Buckingham Palace to receive his knighthood, the exterior shot shows a bare flagpole, which means Queen Elizabeth II is not there. The royal standard flies at any palace where the Queen is present.
Dr. Evil says "I don't speak freaky deaky Dutch," when claiming not to understand Goldmember when he talks about Austin Power's "farjsher" (Father). Dr. Evil was brought up by Belgian parents in Bruges which is a French and Flemish-Dutch speaking city. However, By Dr. Evil's own description of his parents throughout the series, and their questionable sanity and extravagance, it would not be surprising for him to never have been taught common-sense skills while living there.
In the British Intelligence academy graduation flashblack, Austin and Basil Exposition both attended the academy and that they knew each other. But, in the original film, when Basil gives Austin and Mrs. Kensingston the Electric Pussycat Swingers Club assignment, Basil introduces himself to Austin and tells Austin that he works with British Intelligence. Austin and Basil would had never met each other before and did not attend the academy together.
Since Austin's age doesn't change from the freezing process, Nigel should look much older than him, but the age difference doesn't appear to change between the two. Nigel should look 90 rather than 70.
When Austin and Mini-Me fight in Austin's Tokyo hotel room, they both break what are supposedly champagne bottles to create weapons, but you can clearly see there is no liquid coming out when the bottles shatter.
When Nigel, Foxxy, and Austin are chasing Goldmember to the submarine, the Mini's speedometer clearly shows that it is not moving. Also, a warning light on the dash indicates that the rear hatch is open.
The goof items below may give away important plot points.
When Mini-Me and Austin are behind the screen during the physical, the shadow of Mini-Me (who appears to be chewing the "umbilical cord") is not wearing the Austin wig and glasses. When we see behind the screen, he is wearing the wig and glasses.
Near the end of the movie, when the laser is going to be fired, you see several exterior shots of the submarine after it has surfaced. The large window that resembles Dr. Evil's face is completely above water. However, after Nigel's revelation, we see an interior shot showing the submarine to be still underwater (you can see a large fish swimming around on the right side of the giant window).
When Nigel Powers is strapped to the laser beam to have his privates turned gold, Austin and Foxxy are standing on either side of him but the laser is not moving. Also he gets up without his legs being set free.