Contemporary American Poultry - S1-E21
Continuity mistake: When Jeff and Abed are talking in his room, the amount of the root beer float in Abed's glass changes. Especially noticeable is when Jeff leaves, the float goes from near the top and above his fingers to a couple fingers below.
The Science of Illusion - S1-E20
Continuity mistake: When Shirley and Annie chase Jeff, they're wearing their seat belts. In one shot you see Annie doesn't have her seat belt on, but in the next shot it's back on.
Continuity mistake: After the pottery class scene, in the Spanish classroom, when Britta and Jeff are talking, his right arm is across the desk. When Jeff replies "so did someone's psych teacher" and camera cuts, his arm is suddenly folded up by his chest. In the next shot, his left hand is suddenly over his right hand.
Continuity mistake: The "windmill" Pierce draws changes in between shots when the Changs arrive.
Continuity mistake: When Jeff goes into Ian's office and sits down, he puts his book down so the spine is facing the viewer's right and puts his drink on top of it. Later, without touching the book, it is rotated 90°.
Intro to Political Science - S2-E17
Character mistake: When Troy and Abed are covering the school election, on Jeff's detail page, kryptonite is spelled "kyptonite." (00:07:15)
Biology 101 - S3-E1
Visible crew/equipment: At the beginning of the episode, after Pierce rejoins the group and Jeff and the others begin to leave the room, watch Britta as she stands up. When she turns away from the camera, you can see the actress' microphone-pack in the back of her jeans briefly. (It's hard to spot but it's there, and is confirmed in the commentary track.).

Alternative History of the German Invasion - S4-E4
Continuity mistake: When Malcolm McDowell is writing on the board early in the episode, he draws a line under the word "victors". In the next shot of the board, the line has moved, and the gap between the V and I is larger.
Continuity mistake: After Jeff and Britta have had sex, when Britta is pointing her gun at Jeff, in shots behind Jeff she is holding the gun sideways, but in shots from behind Britta she is holding it straight.
Revealing mistake: When Chang's paint bomb explodes, pay attention to the background. You will see paint exploding from three different points on the ground. (One to the center, one to the left and one to the right) The only problem is that there was only one paint bomb in the center, so it makes no sense that there are 3 explosions from different points.
Foosball and Nocturnal Vigilantism - S3-E9
Character mistake: In the bonus clip where we see Leonard's online video about frozen food, the first thing he says is, "This is November 29th". However, if you look under the video, you can see the video was uploaded on Nov 26, 2011, which is three days earlier.
Cooperative Calligraphy - S2-E8
Character mistake: Right before Jeff searches Abed's bag, Britta explains to the group that Abed is the innocent one. After Jeff reminds her that Abed put gum in her hair, she looks at no-one in particular and says "empty the bag, Abed." Jeff is the only one holding the bag, with Abed standing five feet away from both of them. It's clear this line was directed to Jeff, but the wrong name was used.
Continuity mistake: The study-room on the second floor of the library, right next to a set of stairs, and the area isn't particularly bright. In the rest of the series, the study room (which is meant to be the same study-room) is now located on the first floor of the library, nowhere near any visible stairs and the area is much brighter and more well-lit. As often happens, the pilot was filmed before any other episode, and the setting was changed in the meantime.
Contemporary American Poultry - S1-E21
Continuity mistake: At the end when Abed is talking to the Dean, there's an open to-go box with chicken fingers. After the Dean asks who would steal a box of hairnets, it cuts back to Abed and the box and ketchup are gone. Post-it notes are also now seen and the stapler moved. In the next shot of Abed, the chicken is back.
Intermediate Documentary Filmmaking - S2-E16
Visible crew/equipment: When Jeff runs to the car which his father drives and opens the door of the driver, you can see the camera in the side window of the car. You can see it's a set camera and not a 'handheld type', the one that Abed is using during this episode.
Romantic Expressionism - S1-E15
Continuity mistake: At the start of the dance, the Human Being and its partner can be seen dancing on the dance floor, but a few shots later they can be seen entering the room over Jeff's shoulder.
Continuity mistake: In the very first shot, we can see there's a man with a dark coat, sunglasses, short hair and earbuds standing directly in front of the dean. When it cuts to the opposite angle, he's no longer there. When it cuts back to the first angle a few shots later, he's suddenly back.
Asian Population Studies - S2-E12
Continuity mistake: Right after the intro, the group is sitting in the Anthropology class waiting for Prof. Duncan. The chalkboard is filed with writing on it. When Duncan enters, the chalkboard is, except for the 3 top-corner titles, completely clean. (00:03:02)
Grifting 101 - S6-E9
Continuity mistake: When the grifting professor and Jeff are speaking in their office after meeting each other, we see the bottle of alcohol change amounts as the camera moves angles (from Jeff's POV to the teacher's). It is lower when the teacher is all alone, and there's more when it's Jeff's POV. (00:20:14 - 00:20:36)
Audio problem: We hear Troy say "you thought" while he is talking to Annie, but his mouth doesn't move. (00:01:15)
Suggested correction: It's not exploding from 3 points on the ground, it's just 3 different spots on the window. However, the bomb had 3 bottles of paint which is where the three spots come from.
Bishop73
Incorrect. The paint very, comes from three different points in the room. The center of the screen (where Chang is) is where the bulk of it comes from, but there are two extra "bursts" on the left and right side of the screen, far away from where the "paint bomb" is actually located. (And if you don't believe me, look it up on- the scene's on YouTube. The paint burst on the left side of the screen is aimed to the right, and the paint on the right is aimed to the left- which adds a further layer of impossibility to your assertion that the paint all came from the same bomb in the center of the room.