Visible crew/equipment: When Mike and Bobby drive up in the station wagon, just as Mike stops the car, on the left side of the screen we can see the edge of the house set. Instead of the usual wood fence in the yard, there appears to be a directors chair (provided to all cast members) in the dark background. (00:05:35)
Plot hole: When Mike was at the meeting and pulled out the Yogi Bear poster, he said he could find the other cylinder, and was told by one of the other men that they had to catch a plane in 30 minutes. After that, he had to find Jan, and then Jan and Marcia had time to go on about 4 rides and then find the sketches and then start their run across the park. There is no way all of that could have been done in less than 30 minutes.
Confessions, Confessions - S2-E12
Revealing mistake: At the start of dinner, in the wide shots we can see the table and chairs directly on the carpet. When Carol notices the leaky vase and calls for Alice, in the wide shot as Alice approaches the table, now there's plastic sheeting spread across the carpet under the table, as well some visible tubing under the table, presumably for the water stunt. (00:05:45 - 00:08:15)
The Impractical Joker - S2-E13
Continuity mistake: While Myron is on the closet shelf with Carol's shoes, the camel colored shoes repeatedly change between having camel colored heels to dark brown heels. (00:11:35)
The Voice of Christmas - S1-E12
Character mistake: Alice is in Mike/Carol's bedroom trying to find a place to hide a present and can't find room in any closets and chooses under the bed (only to find Bobby there first with the same idea). Why couldn't Alice simply hide the present in her own room?
To Move or Not to Move - S1-E23
Plot hole: The "unexplained noises" (the strange moaning) heard by Carol and Alice could not be heard so loud and clear from the room they are standing in (like we hear in the show). Greg later admitted the noises were on a tape recorder, which he had to have operated, and since he wasn't in their room, the sound would be lower and from another room in the house. Definitely not like how it appeared to them in the show.
Suggested correction: Greg could have used an extension cord, left the tape recorder in one room and plugged it in to play when he wanted to. That's what I did to my younger sister to make her think a monster lived under her bed; she heard it "breathing" when she was alone in the room.
That partially makes sense. Greg could not control when to start the tape because he has to be in the room to hit play. So the only way this could possibly be correct is if he played the tape before Carol and Alice entered the room, and he recorded it with several minutes of dead air before the noises came on, and left it running the entire time.
Only problem with this theory is, Carol and Alice heard it from at least two different rooms in the house at different times. How could Greg have set up the recorder and played it in advance, before knowing what room they'd be in each time?
Character mistake: While they're dancing and practising in the garage Marsha says "come on Chris", when she should have said Peter.
Is There a Doctor in the House? - S1-E13
Other mistake: At the end of the episode, when they were looking at the chart they created, and Alice was found to add the measles, you see Mike Brady adding the word "Alice." But the kids and Tiger all had columns, but he is adding her as an extra row, which was for the different illnesses, to suggest that Alice was an illness instead of a person having the illness.
Continuity mistake: The home number Mike dials in the men's bathroom to check on the kids, is very different than the home number Carol dials in the ladies restroom. (00:18:30 - 00:18:56)
Visible crew/equipment: A shadow of a boom mic is visible moving across the brick wall just as Alice starts to punch Mike in the stomach.
Audio problem: When Peter is hinting to Carol that he is still growing, the audio changes to being dubbed in from Peter separate from the actual scene after Peter stands up and mentions he wants cowboy boots. Then the audio is back to normal once Peter mentions "Brown" cowboy boots.
Continuity mistake: The kids walk into the kitchen complaining about the noise from Jan's tap dancing. Peter reaches up and grabs his paper sack lunch and then in the next shot he says "talk about noise pollution" while grabbing his lunch for a second time. (00:09:07)
To Move or Not to Move - S1-E23
Visible crew/equipment: When Alice and some of the kids are in the den Bobby gets up to turn off the TV, and the shadow of the moving boom mic is visible on the wall behind Peter. (00:14:05)
Continuity mistake: When Mike walks into the kitchen asking Alice if he can give her a hand, Alice removes two pots from the stovetop leaving the left side clear, but in the shots from the dining room when Alice trips, there are two large pots on the left side of the stovetop, near the edge of the counter. (00:01:20)
The Grass is Always Greener - S1-E24
Visible crew/equipment: When Mike returns home with the grocery bags, as he sets them on the table Alice walks over, and the shadow of the moving boom mic is visible on the swing door behind Mike, at the top left side of the screen. (00:11:05)
Visible crew/equipment: While Mike is fiddling with his golf bag the morning after Tiger disappeared, just as Mike tells Bobby there's nothing to worry about because his siblings are checking the neighborhood, the moving shadow of the boom mic is visible in the top right corner of the screen. (00:02:45)
Continuity mistake: In a first-season episode, Carol is asked by one of her daughters, "How come you never play golf anymore?" indicating she used to play golf. In the later seasons, an episode came out where Carol was first learning to play golf (appearing as if she's never played before).
Dough Re Mi - S3-E16
Plot hole: Greg bought a time spot in the music studio to record a song. He writes a sudden new song for the studio at the last minute and then, next scene, the kids are singing the song in the studio along with background music playing. Where did all the instruments come from and who was playing them? Even if pre-recorded, Greg and the other kids would have just shown up at the studio at their time, and no way they would have been able to get the tracks down in advance to an unknown song like that.
Fright Night - S4-E6
Plot hole: After the girls get their revenge on the boys in the attic, they explain the mysterious "I must have air" voice from the trunk as it's revealed to be from a tape recorder. Marcia mentions a set up the girls had where they played the tape recording (from their bedroom) when they were ready. This is impossible as the recorder was in the attic trunk and the play button would have had to be manually pushed down by an actual person to activate.
Character mistake: At the end of the episode when all the kids line up to head upstairs, Mike gives instructions to line up according to their size (big to small). If you notice carefully when the kids all line up, Peter stands behind Marcia, yet he is slightly taller than her.