Deliberate mistake: When the Doctor begins to "fade" in the transporter room his mobile emitter fades with him. Since it's made of solid matter and is not a hologram, this shouldn't be possible.
The Builders - S1-E2
Deliberate mistake: When Basil is arguing with Polly and eventually decides to call O'Reilly, he trips over the gnome. However when we see Basil come out from behind the desk, he takes the gnome out of a side gap in the desk, perfectly out of reach of Basil tripping over it. (00:14:30)
Deliberate mistake: The show is set in New York. Joey was a cast member on Days of Our Lives, which is filmed in Los Angeles, but is still always around.
Tabula Rasa - S1-E3
Deliberate mistake: As Ray drives the pickup truck, an oncoming vehicle beeps and the marshal heads behind Ray's pickup. In the shot through the cab's rear window facing the bed, as the marshal pulls behind, the bed's side wall interior is seen (note the metal loop tie down on top) and the entire tailgate is gone; yet the tailgate is intact later, when the pickup flips over. The tailgate was removed for the shot to allow visibility of the marshal's vehicle.
Deliberate mistake: After Barney buys the car from Mrs. Lesh, he and Andy are standing at the steps as he tells her not to look back at the car, and the next shot of Andy and Barney at the front door is the same shot from a few moments prior, but it's been flipped so they would be looking in the opposite direction. (00:09:30)
Deliberate mistake: Robert Brown wears a gold turtleneck sweater with his leather outfit throughout the series. Though it's more than 100 years beyond the show's 19th Century setting, he and the producers decided that he looked good in it, so allowed the anachronism to stand.
Mr. Monk Meets His Dad - S5-E9
Deliberate mistake: In the opening scene, Wood's tie is grabbed by an exposed fan belt located above the side of the engine. However, this belt/pulley assembly doesn't exist in a real truck's engine compartment - the fan belts are all in front of the engine, where the camera can't see them easily. The prop master must have added this dangerous device to the truck to allow the camera to see the strangulation.
Deliberate mistake: The final Auton attack in this episode is supposed to take place in a busy London street, but it is a pedestrianised area, and several shots reveal this. For example, there are no road markings visible, no apparent distinction between pavement and road, and a bus stop appears to sit in the middle of the 'road'.
Deliberate mistake: Every time the commandos are seen in the early episodes of this season (The Freshman and Fear, Itself) they wear ski masks to protect their identities. Starting in The Initiative, they stop doing this. The reason is that the audience needs to be able to identify the different commando characters, but it's still a discontinuity.
Deliberate mistake: Special effects of the ship in space were very expensive in the 60s and couldn't be wasted. So when a larger model was built with slightly different nacelles, shots of both versions became common, even within the same episode. This is why the Enterprise sometimes had red needle-tipped nacelles and sometimes lighted "spinning" ones, and in aft views she had either round white spheres or perforated vents at the nacelles' ends.
Deliberate mistake: In season 4 episode 7, 'Brian the Bachelor'. Cleveland is sitting in a chair in a waiting room and is getting really nervous. Peter tries to help him by tying him to a chair and when that doesn't work, he takes off all of their clothes. Peter ties Cleveland's hands to the arms of the chair and then takes Cleveland's clothes off. He couldn't have taken the shirt off while Cleveland's hands were tied to something without untying him or ripping the shirt, which he didn't do.
Twenty-something girls vs. thirty-something women - S2-E17
Deliberate mistake: In the scene at the beach party, when Carrie brings everyone beers, Miranda says "There's so much foam, I could ski on the head," but the cups are tipped toward camera and you can see there's no foam at all.
Boys in the Band / Hex Games - S2-E2
Deliberate mistake: When AJ and Chester show Timmy "The Timmy Tuck", AJ says it has a 1400 degree turn. The diagram shows 140°.
Deliberate mistake: Coming back to Earth in 1998, Stone is amazed to see the Cincinnati Reds and the NY Yankees on TV in an interleague game. This wasn't yet possible in '98, as an American League East team could then play only a National League East team (and the Reds are NL Central). But the producers put it in anyway as their own little "fantasy baseball game," because these were their favorite teams.
Deliberate mistake: Every time there is a scene on the beach, the background is so fake - the waves never move.
Deliberate mistake: The dying Vantika hides his consciousness away in a portion of Dr. Bashir's mind. At one point, "Bashir" (heavily cloaked in clothing and shadows) comes up behind Quark and makes sure his demands are clear. At this point, it should still be unclear just how Vantika is still "alive", and whose body he is using. However, when he shoves Quark away, for a fraction of a second, Bashir's face can be seen under his hood. They were true to plot in using the right actor (though he shouldn't have been seen), it is possible some might have spied the secret earlier than the show meant to reveal it.
The Anniversary Present - S2-E1
Deliberate mistake: In this episode Lucy is trying to get Ricky to notice that their anniversary is on Monday the 19th by showing him a large calendar with the 19th circled. However, when they are in California and they celebrate their anniversary and Ricky asks for information to be wired to him, the telegram tells him that their anniversary is on the seventh.
Deliberate mistake: At the beginning, Asajj Ventress is wearing an outfit with a long skirt. When she is rescued from her damaged starfighter by the scavengers, the skirt has vanished. This was done because the animators found the skirt too difficult to animate. This is also why she receives a brand new outfit that never had a long skirt later in the episode. (00:08:20)
Deliberate mistake: While Bobby's parakeet is flying around downstairs, before the bird settles down on the TV set there are two closeups of Bobby which are flipped shots, note the furniture and Bobby's PJs. (00:07:00)
Deliberate mistake: When Johnny is watching the news footage of the Hulk battling Thing, all the footage is merely clips from that portion of the episode, including parts of the battle that were happening before the camera crews even showed up. (00:02:50 - 00:07:20)