The remote control device responding to the homing beacon on Fat Bastard shows a tube map with the (pink) Hammersmith & City line showing, except that the line was considered part of the (red/purple) Metropolitan line until the late 1980s. The map we see is supposedly in 1969, and thus the tube map would have looked very different.
In the 1969 era scene with the President, there is a US Army 4 star general wearing the dress green uniform. The actor is wearing a light green shirt. The US Army wore khaki shirts with the dress green uniforms until 1983 when they changed to the current light green shirt.
While singing his "Just the Two of Us" duet remix with Mini-Me, Dr. Evil states that Frau revealed Mini-Me's existence to him. However, Number Two introduces him for the first time in his Starbuck's lair.
When Austin Powers and Ivana Humpalot are playing chess, the board is set up wrong, since we can clearly see that a black square is on the right. A white square is always on the right for both players.
When greeting Frau Farbissina, Dr. Evil asks, "Wie geht's es Ihnen?" which is supposed to mean literally "How is it going for you?" in formal German. However, he speaks redundantly, as "geht's" is a contraction of "geht es".
When Austin travels back to 1969 in the hangar, and backs up into the rear rack of computer monitors, he knocks the entire thing to the ground. But in later shots, the rack re-appears upright and then is seen on the ground again.
When Felicity is hammering the tennis racket into Austin's "shadow", it is almost up to the strings when the camera moves to the guards outside the tent. When the camera turns back to the shadow, it is only "in" up to the middle of the handle.
When Scott travels back to 1969 and arrives at the Volcano Lair, he tells Dr. Evil "He'll (Mini-Me) kill me the first chance he gets." In this over-the-shoulder shot, Dr. Evil is looking at Scott. In the next shot when Dr. Evil says "Probably," he turns to Scott, when he should have already been looking at him.
While Austin and Felicity are dancing to "I'll Never Fall In Love Again", there is a girl in yellow standing behind them. In the next shot, the same girl walks past them from a completely different place.
When Austin and Robin Spitz Swallows are being shot at with the machine gun, Austin and Robin walk right, then in the next shot they are walking in the opposite direction, but the gun man is walking the same way the whole time.
While Felicity and Austin are dancing, there is a group of people standing near a garbage can on the street corner watching them. The next time we see the garbage can they are gone, but then they reappear later.
After Dr. Evil fights the guy who says "What are you, a freak?", he calms down. Before he goes to fight him again, you can see Scott to the right of him. He is not being controlled by the two "Jerry Springer crew." In the next shot of Scott, he is. Then he's not being controlled in the next shot.
In Starbucks, when Frau introduces her girlfriend Unibrau, there are two Starbucks "baristas" working behind the counter in the background and one walking in front of the counter. When we cut to a close-up of Unibrau, the baristas vanish.
When Mini-Me drives the 3 wheel motorized bicycle down the ramp in Dr. Evil's volcano lair, the Burroughs B-205 tape drive on the right side of the Sage computer panels, at the bottom of the ramp, is missing. In the next scene the tape drive is present.
When Austin and Felicity are in the cell, overhead shots show them moving closer to the walls, due to the floor splitting open. However, in close-ups of their faces, the background behind them doesn't come any closer.
Several actors are playing the same characters they played in the previous movie, yet they have different names here. Clint Howard's radar control character was named "Ritter" in the first film, yet it's changed to Johnson here, and even more confusingly, the credits call his character "Peters." Likewise, Charles Napier played "Commander Gilmour" in the first movie, but the end credits here call him "Hawk." (The scenes are set in the past so it's possible these are parents/distant relatives of the characters from the present, but if that's the case they'd likely have the same surnames.)
When Austin Powers is fighting with Mini-Me, at one point, Mini-Me picks up Austin and twirls around holding Austin over his head. When it switches to the close up, Mini-Me Is standing still, using just his arms. When the perspective changes back to the full scene, Mini-me is spinning his whole body again.
In Austin Powers: International Man of Mystery, Basil sports a blond 60's-style hairdo in the 60's. In this movie, when Basil is seen in 1969, he has the same hairstyle and color as he does in the 1990's.
When Austin is trying to "flush" Mini-me from his pants on the moon base, he is reading a newspaper. The name of the paper indicates it is from London, however the back of the last page clearly shows a USA weather map.
In the darkroom, Austin makes 8X10's of the pictures he took earlier. However, the door is left open, so the light coming from outside would totally ruin the photos. Since the photos weren't washed, they would deteriorate much faster over time.
When Austin crashes through the window with Robin Spitz Swallows and Felicity gives him a ride away in her car you can clearly see a British Cab with a white license plate. The plates should be yellow at the back and white at the front.
When Austin goes back in time to save Felicity on the moon, he inputs his destination as "ten minutes ago." But when Austin is running down the stairs to get to the time portal the screen that shows the destination already has "ten minutes ago" on the screen, before Austin ever types it in.
During Austin Powers' time travel, his car enters the building only a few stories up. Austin never changes floors throughout his entire visit in the nightclub, yet when he and Robin Spitz Swallows crash through a window, they fall much more than just a few stories, as if they were falling off of a skyscraper. Given the over-the-top nature of this particular scene, this is probably deliberate.
Just before Burt Bacharach and Elvis Costello sing, the camera pans. You can clearly see the mountains of LA/California in the background, although the scene is set in London.
Earlier, Austin says, it was remarkable how England looked "in no way like southern California", ironically speaking. Showing Californian mountains in the background of London therefore is merely consistent.
Elvis Costello appears as an adult in a scene set in 1969, but he was only fifteen years old that year. However, both the good and bad sides have access to time travel, so the modern-day Costello probably came back in time to perform for Austin.
As Mini-Me tried to kill Scott Evil by pressing the red button on the control panel, Mini-Me's name already appears as one of the other four choices. About a minute ago, Dr. Evil didn't even know he had a miniature clone of himself, and this clone has not yet been named by anyone.
In the previous film, Number Two betrayed Dr. Evil and was incinerated. He obviously survived (much like Mustafa did; and judging from the burn on his face) but it is very unclear how and why he became faithful to Dr. Evil again. He didn't fully support Dr. Evil before his betrayal and Dr. Evil tried to kill him, hence this being one of the biggest plot holes between the two films.
Since the series specializes in spoofing the real history of the "past" that they travel to, the 1969 President should be a caricature of Richard Nixon. Yet there is no clue that this is supposed to be Nixon.
When Dr. Evil is saying that he has a mole in the Ministry of Defense to steal Austin Powers' mojo (after the first "really" line, at the 22.15 mark), Young Number Two's healthy right eye is visible under the patch.
The goof items below may give away important plot points.
After Dr. Evil recovers Mini Me from space, he says he is giving Mini Me a shot because he is concerned he will get sick from the cold of space. While cold temperatures do make one more vulnerable to viruses, in the vacuum of space most viruses cannot survive and even if some did, they would be so literally astronomically small in number as be effectively zero.
Robin Spitz Swallows is shot in the back several times with a machine gun causing holes to be put in her coat she is wearing. When they fall out of the window, the holes cannot be seen on her back as they once were the last shot.