Character mistake: Austin is supposed to be the archetypal Swinging Sixties Londoner yet he refers to his car as a Shaguar, a pun on 'Jaguar', pronouncing it 'Shag-war'. A native of this green and pleasant land would pronounce it 'Shag-u-ar', three syllables, not two, based on our pronunciation of the car manufacturer's name: Jag-u-ar.
Revealing mistake: In the scene where Austin and his allies burst through the big blue door, right before "Secret Agent Man" begins playing, the door explodes, and you can clearly see that it's made out of wood, even though it's painted to look like and is meant to be metal. This mistake is even mentioned in the director's commentary on the Austin Powers DVD.
Continuity mistake: In the one scene in Dr. Evil's evil headquarters, it shows the large screen opening in front of them. The screen shows the warhead that they're going to arm the drill with. The table in the room is reflective, it shows the panels of the screen opening, but it does not show the image that's on the screen reflected in the table.
Continuity mistake: Towards the end of the movie Dr. Evil is talking to Austin in his evil headquarters. Dr. Evil is sitting in a chair with Mr. Biggelsworth in his arms. The scene has close ups of Dr. Evil with a sleeping cat, and broad shots of Austin and Dr. Evil, with the cat trying to get away. It keeps switching from one shot to the other.