The Habitation Configuration - S6-E7
Continuity mistake: When Howard is almost done moving out of his mother's house, there is a table in front of him with a measuring device in the top corner. In one shot, the device moves to the opposite corner of the table. In a few shots later, the item returns to its original position.
The Separation Oscillation - S9-E2
Continuity mistake: Sheldon is returning Amy's scarf and says, "You wore it the night we went ice skating, remember?" to which Amy replies, "You mean the night that I went ice skating and you stood at the rail Googling the symptoms of hypothermia." But in episode 8-12: The Space Probe Disintegration, Amy states that she cannot go ice skating because she has "unnaturally brittle ankles." (00:04:55)
The First Pitch Insufficiency - S8-E3
Continuity mistake: When Howard is on the mound, the shot from behind the plate shows a player ready to catch the ball in a uniform shirt. The next long shot from the stands shows a catcher in full gear, and then back to behind the plate to the player in uniform shirt only.
The Discovery Dissipation - S7-E10
Continuity mistake: The Stevenson Award in Sheldon's office reads "Sheldon Cooper, PhD." But, in the "Dennis Kim" episode we are told Sheldon was 14.5 years old when he earned that award - 1.5 years before earning his PhD at age 16, so the award should not show the title of PhD.
The Skank Reflex Analysis - S5-E1
Continuity mistake: At the start of the episode, Raj's Sirkist can rotates itself on the table.
The Fuzzy Boots Corollary - S1-E3
Continuity mistake: When Sheldon is telling Leonard that he didn't ask Penny out, there is a yellow food-bag on the counter between them that keeps changing its position with every shot change. (00:11:15)
The Hamburger Postulate - S1-E5
Continuity mistake: At the end when Sheldon has got a burger, the amount of burger left changes depending on the camera angle. Most noticeable when he asks Penny about permanently reserving the table - between shots the lettuce disappears, the bun changes colour, and the bite marks change.
The Boyfriend Complexity - S4-E9
Continuity mistake: In 2-15 "The Maternal Capacitance" Penny explains to Beverly, referring to her father, "My mom could have just said 'Bob, get over it, she's a girl'". But in this episode Penny's father is named Wyatt.
The Good Guy Fluctuation - S5-E7
Continuity mistake: Leonard refuses Penny's offer of a glass of wine because the sulphates in wine give him migraines. This makes no sense - we have seen him drinking wine in too many episodes to count without him once complaining of an adverse reaction.
Continuity mistake: In the laundry room when Penny is talking to Sheldon, Sheldon takes some time arranging a blue shirt on his plastic laundry folder. Then it suddenly turns into a pair of socks, which he folds and then picks up the blue shirt again. There was not enough time for him to switch out the shirt for the socks. (00:06:30)
The Long Distance Dissonance - S10-E24
Continuity mistake: After the 'Ramona' conversation between Sheldon and Penny ends, Sheldon leaves Penny alone on the couch. Penny looks up to the "Amy and Penny" painting and says "Don't look at me like that, I tried." In S10 E10, Amy gave the painting back to Penny, which was hung in Leonard's living room.
The Long Distance Dissonance - S10-E24
Continuity mistake: When Sheldon takes a plane to see Amy in Princeton, the plane he is on has 2 engines when it takes off and 4 engines when it lands. It's a direct flight from LAX so he wouldn't have changed planes.
Continuity mistake: Penny throws a black iPod out the window, but when Raj comes in having found it on the ground outside he has a silver one. (00:01:00 - 00:02:45)
The Agreement Dissection - S4-E21
Continuity mistake: When the girls take Sheldon out dancing, he mentions that he considers himself a proficient dancer due to his Cotillion training when he was a boy. But in S1E7 The Dumpling Paradox, when Leonard suggests that they could have gone dancing with the girls, Sheldon said that nobody in the group - including him - could dance.
The Platonic Permutation - S9-E9
Continuity mistake: In episode 16 of season 1, Penny is talking about making a cake for Leonard's birthday and states she knows his birthday because she was checking a horoscope for him. But in this episode she does not know when his birthday is, which leads to Leonard revealing he was secretly reading Penny's diary. (00:40:00)
Suggested correction: It's certainly possible to forget something you knew eight years ago.
This is perhaps true for two people who rarely have contact, but I would list this as a character mistake. Penny and Leonard are in a serious relationship and are married at this point.
This is also a couple who BOTH forgot it was their anniversary.
I forgot my wife's birthday on a number of occasions. It's one of the reasons she is now my ex-wife.
There's a difference between forgetting and not knowing.
Continuity mistake: Sheldon puts the bag in the bin at the start of the episode so that the handles are facing the ground. When he gets the bag out, the handles are at the top of the bin.
The Vartabedian Conundrum - S2-E10
Continuity mistake: When Leonard and Penny are talking in the dryer room, the size of the towel Penny is folding keeps changing. (00:15:00)
The Shiny Trinket Maneuver - S5-E12
Continuity mistake: Howard, Leonard and Raj play Jenga in a strange sort of way. Most notable is the playing order, 1. Howard, 2. Raj, 3. Raj, 4. Leonard, 5. Howard. When it's Leonard's turn he doesn't move any bricks, only pokes the tower and it's then Howard's turn. When its Raj's first turn he puts a brick on the top of the tower. When it's his turn again, the brick is gone and he puts a new one in its place.