Earlier skits with the yip-yip martians have them consulting an "Earth book" to learn about different things they discover on Earth. Yet a later skit has them discovering a book for the first time and trying to figure out what you do with a book.
In skits discussing how many sides certain shapes have, some say a circle has one side while others say it doesn't have any sides.
When Cookie Monster was a contestant on The Remembering Game, in worry of him winning the jet plane instead of a cookie, he says, "No can eat jet plane!" Cookie has eaten many otherwise inedible items in the span of the show, especially if he either couldn't get cookies or was out of cookies (like a pickup truck and Hooper's Store). Therefore, if he had kept the jet, he would've eaten it somehow.
In Bert's song, "I'm Square", he says he likes going to the beach when it's cloudy. Many skits before and after proves this is not true, especially the skit where he came all prepared and Ernie wound up bringing only his umbrella, and when it started raining, Bert started freaking out.
In the skit where we learn how crayons are made, the girl who decides to look at her orange crayon with a sense of wonder is holding up an obviously lightly used crayon. However, at the end when she returns it to the box, it's brand new and unused.
During the "Beat the Time" skit with The Count, Guy Smiley declared that The Count won when he created his thunder and lightning from counting the 20 seconds he had, as he had found two thing that came from the sky. However, the thunder and lightning came after the time ran out, so technically he still lost.
One skit involving a little girl in a picture frame, in limited animation, showing off different hats. It has her showing off a cowboy hat, a clown/circus hat, and a straw/fruit hat. However she also shows off the headgear worn by American football players and firefighters, and says they're hats. Those are both helmets.
In a lot of times you can see the strings of the puppets that are being used by the puppeteers.
One skit involved Cookie Monster about to eat a cookie in his hand, but then a girl Muppet comes around with music playing, making him compelled to share his cookie with her. After Cookie devours his half as he normally does, the girl devours hers in the same manner. However, you can clearly see her half of the cookie drop out of her hand before it goes into her mouth. (This may have been intentional as she was not a puppet designed to "eat" anything like Cookie was.)
In the skit involving Cookie Monster at a carnival and trying out the "ring the bell" test of strength game, when he realizes the prize was a cookie, and it compels him to hit the platform hard enough to not only ring the bell but destroy the game, it is clear as day that the weight never goes past 5 and the game just shakes apart.
The live action stop-motion skit showing a dune buggy being built ends with it driving away, but there's nobody in the driver's seat.
In the cartoon skit of a mother and child deciding to make a snack of homemade popsicles. The straws used for the handles on the popsicles were just put in the cups slanted in all different angles (without using tinfoil or something to hold them straight), but once they were frozen and removed from the freezer (with the straws still slanted), they were all straight when removed from their respective cups.
A skit involving The Fonz from Happy Days doing his usual "Ayyyyy!" towards the camera, and the letter "A" appearing below him, and him looking down at it. Later, the A was shifted to the left of him, but he still looks down.
Often, a Muppet will start speaking but their mouth opens a split second after they start.
During the final stanza of the Anything Muppets' song "J Friends", when the four Muppets jump up at the line "Let's jump with Jane", the hair and forehead of Muppet performer Frank Oz are briefly visible at the bottom of the screen.
In the end of the song "Hey Food" The drummer's drum falls off the set.
During the "Remembering Game" sketch, when Cookie Monster calls "Number 4", a stagehand's arm is visible reaching behind the game board at the bottom right of the screen.
In the bedroom skit, where Ernie woke up and was determined to find out what time it was, he couldn't see the clock nor see the light switch on the wall. The clock, maybe (as it was an old school analog clock that wasn't back-lit or anything), but for the latter, in every other bedroom skit, the lamp on the nightstand in between them was always able to be turned on and off right there, and even if they did use the light switch, Ernie would know exactly where it is, especially in limited darkness.