Visible crew/equipment: When Dewey yells for Lois because the bow was put on the present, if you look to the left of Lois, you will see a crewmember kneeling down, handing Lois the Red Dress, who is holding a bucket of water. He quickly gets out of the camera's view. [This is on the DVD of the first season. Someone said on the audio commentary that after catching it, they deliberately left it in to see if anyone would spot it. It's still a mistake, but that's why it's there.]
Visible crew/equipment: There is a man's elbow on the right of the screen near the start when only Jamie and Hal are in the room. (00:02:49)
Plot hole: Hal says that his father was never there when he got dumped. But in episode 7, "Francis Escapes" Hal tells Francis he's never been dumped.
Visible crew/equipment: When Hal is walking the stroller with Jamie inside at the park and Hal approaches the Body Builders, the full body of a camera-man and the camera he's filming with is ridiculously visible on the right side of the screen in a wide-shot.
Home Alone 4 - S1-E3
Visible crew/equipment: While Malcolm, Reese, Dewey, and Francis are deliberately making their house messy, you can see right above their ceiling a bright light shining down and a black background. It shows that they film their interior of their house on a set.
Continuity mistake: When Lois is yelling at the boys for playing with fire, the location of the green scissors on the table keeps changing between shots.
Visible crew/equipment: When Malcolm is first visiting Stevie, and Stevie is telling the joke, you can see the hair and hand of a crew member coaching Stevie through the joke.
Continuity mistake: In the scene where the grandma is talking at the table with Reese and Malcolm, notice the bottle on the table next the the grandma, between takes the cork from the bottle goes from being on top of the bottle to next to the bottle, it changes in almost every different shot.
Experiment - S5-E18
Revealing mistake: In the first scene, Lois is feeding Jamie while talking to Dewey. In the next clip, it changes to just a prop baby doll toy with a beanie covering a majority of his face. (00:04:25)
Stupid Girl - S4-E4
Visible crew/equipment: When Hal is driving the steamroller at night alone in the car park - just before Dewey shows up - in the first shot there is a crewmen standing smiling in shorts to the side of the roller who then slowly walks out of view.
Visible crew/equipment: After the theme song ends it starts with Lois just finishing shaving Hal's back. Off to the side you can see the crew kneeling down.
Visible crew/equipment: When Hal rebuilds the track of dominoes, he walks out and slams the the door. This starts the dominoes falling. When it comes up to a shot of the American flag in dominoes, if you look closely to the left of the screen you can see a hand pushing it over.
Audio problem: Near the end when the family have got back in the car Malcolm says something like "who would have thought a traffic jam could be so cool, we should do this again". Near the end of the sentence we can see Malcolm's mouth and it isn't moving.
Revealing mistake: In an imagination sequence Reese and Malcolm are throwing around Jamie, but you can see it is a doll.
Continuity mistake: When Malcolm is trying to get the TV back to the channel he was watching, he picks up a big remote and finds the channel which the video camera feed is on, but the remote he is holding on the TV is a smaller one. It then cuts back to Malcolm who is moving his hand away from the remote he just put down and it's the bigger one again.
Continuity mistake: At school, Dewey stops the shouting match between the two opponents. In the shots facing Dewey, the boy's arms are held way up high in front of him, but in the shots facing the kids, the boy's arms are down at his sides.
Answer: Francis is only allowed to talk on the phone at school for three minutes a day. I think the code gives him free access to talk to his girlfriend.