The song "You Are My Sunshine" by Jimmie Davis was not written until 1939 and not recorded and released until 1940. The calendar at the radio station puts the year at 1937, three years too early. (The filmmakers knew this anyway)
There is a very heavy focus on the use of the Confederate Battle Flag at the KKK rally. However, the association of the KKK (and racists in general) with the "Rebel" flag grew out of the Civil Rights conflict of the 1960s. During the Twenties and Thirties, the peak of KKK membership, only the U.S. flag was represented at KKK rallies, even in Mississippi.
When Everett and gang enter the radio station, he asks, "Who's the honcho around here?" The word "honcho" is taken from the Japanese 'Hancho', which means "group leader," and did not become an English expression until GI's brought it back from the Pacific war. Its first recorded use in the U.S. is in 1947, many years after this movie's timeframe.
When Everett and Delmar are eating at the restaurant, Everett orders the restaurant's "finest bottle of bubbly wine." Though Prohibition was repealed nationally in 1933, Mississippi still prohibited the sale of alcohol until 1966.
When the Soggy Bottom Boys perform onstage with beards, lighting fixtures can be seen from behind. These lighting instruments are ERS type and the kind shown weren't invented until the late-1980s, and couldn't have appeared during the time era of the movie.
Baby Face Nelson asks whether Ulysses, Pete, and Delmar know their way around a Walther PPK. While the PPK was produced in Germany starting in 1929, it didn't gain a market in the US until after WWII, and sales only really took off after the introduction of the James Bond movies in 1962. Bank robbers of the 1920s and '30s generally preferred American-made pistols, like the Browning 1903 or the Colt 1911.
In the movie theater (and elsewhere) the guards are shown holding Remington 870 shotguns, which were introduced in the early '50s. It would be more accurate to show them holding Winchester Model 12s or maybe Ithaca 37s.
During the Soggy Bottom Boys' performance near the end, the bassist can clearly be shown playing an all metal set of strings. Metal strings for an upright bass were not available until the '60s. The correct material would be gut (more commonly known as catgut) which is brown in color and slightly translucent. Gut strings are very important to the sound of traditional bluegrass and country music.
The group singing "Keep on the Sunny Side" at the rally is prominently accompanied by a Dobro (a resonator guitar) being played with a slide... but there is none to be seen. There are other people on stage, but it becomes clear that none of them has an instrument.
During the café scene, Big Dan's lips don't match his words when he's talking about how he sells Bibles. You can see this clearly when the camera faces Everett, just as Big Dan says, "From Genesis on down to the book of Revelation."
Pete says "today is the seventeenth" and that they have only four days to reach their goal before it is flooded. However, there are at least six different night scenes from that point until the scene in which the flooding takes place, and the newspaper Everett throws on the fire has the date "July 13, 1937" and mentions the Soggy Bottom Boys.
When Ulysses, Pete and Delmar wind up their recording of "Man of Constant Sorrow", but before the guitar fades out, Ulysses lets out a loud whoop. Since the recording was direct to disk, this would have effectively ruined the ending of the recording and necessitated another take.
When the guys come out of the recording studio after recording "Man of Constant Sorrow" Delmar is covered in a white powder, but when they go in and while recording the song he is not. Also, while they boys are walking down the road before Baby Face Nelson picks them up, Delmar is again covered in the same white powder, but is not after they get into the car.
When the car breaks down, Everett asks the store clerk for a replacement part. The clerk tells Everett that it will take two weeks to order the part. Everett tells the clerk that two weeks is too long, and he returns to meet Pete and Delmar and tell them the bad news. Everett and Pete then argue over the pocket-watch that was stolen, and they are interrupted by the river baptism. After the baptism, the three men are seen driving away in the car without any explanation about how it was repaired.
When the car breaks down following their escape from the burning shed and they tell the kid to go on back home to his Pa, he walks off in the direction the car was going which would of course been the opposite direction from where the boy lived.
After the flood, Everett, Delmar, and Pete find themselves drifting on the coffin. You can tell that they are either are being pulled/pushed up stream or they are not moving at all; the debris and broken trees are floating downstream pass them as they converse.
When Delmar and Ulysses help Pete escape from prison again, Pete is in traditional prison garb. Later, they find the Klan rally and steal the Color Guards' clothes. After that, at the Soggy Bottom boys' performance, he is wearing new clothes. Although these clothes could have been from the Color Guard, it doesn't seem like he would have enough time to put them on before saving Tommy, and Delmar and Ulysses didn't seem to be carrying extra clothes on them.
During his argument with his son about the campaign, Pappy's dinner plate changes positions several times. At times it is directly in front of him, while other times it is clearly off to his left side.
When the boys pull up to the radio station to record their song, the car is parked parallel to the building. In the following scene, when they get out of the car, it is parked facing away from the building.
In the overhead shot of people walking into the river from behind them, the preacher and the person being baptized are deep in a tree shadow. When Delmar emerges some the same place and declares that the "preacher has washed his sins away", the shadow has moved several feet away from the preacher.
When Big Dan (John Goodman) beats up Delmar (Tim Blake Nelson) the blood on Delmar's face changes. It is already dried up and reaches down to his jaw just a few seconds later when the camera shows Delmar on the ground. A few seconds later the blood is a bit wet and spread out more above the lip. After Big Dan passes him, the blood is dried up again and only reaches to his lip, not his jaw.
When Pappy O'Daniel is yelling at his son on the porch about his suggestion of getting a smaller midget than Homer Stokes', his head is turned towards his son, in the shot immediately after his head is facing forward.
When Baby Face Nelson enters the bank to rob it, his face is clean. After he walks past the crowd, there is a smear of lipstick across the left corner of his mouth. There is a woman in a peach dress that steps back from him as if he recently released her. Probably edited out a short scene of him grabbing and kissing her, before walking up to the counter to get the money. The lipstick remains on his face into the following scene around the campfire.
The scenes before and after the flood must only be one or two minutes apart, as they were able to hold their breath and not drown. But before the flood it is very sunny (strong shadows), and after the flood the sky is a hazy white (no shadows).
When Delmar runs to be baptized, he cuts in front of a bald man at the front of the left-hand line. The shot changes, and when it returns, the bald man has disappeared as Delmar is being baptized, and the two women who were second and third in line have moved to the front.
When the blind man arrives following the passing train the caboose can be seen on the tracks just ahead. After the three escapees jump on the moving handcar the camera pans to the track ahead and no train can be seen.
When Pete is being whipped for information, one of the posse members throws a rope and noose over a naked tree-branch. The noose then dangles in Pete's face. But there should already be another rope over that branch, as Pete is clearly being suspended by his tied hands, and that's the only overhead spot where it could be attached.
When Everett and the boys arrive at the town to find his daughters singing in the election rally, his moustache is visible when he is talking to Pete and Delmar, but when he goes over to talk to his daughters it disappears, and then reappears in the following scene
When Pappy is talking with his employees after Big Dan, Everett, and Delmar leave the restaurant, the orange on his plate changes from shot to shot; it goes from being cut cleanly on the top to being cut serrated-style.
When Pappy O'Daniel and his staffer, who refers to him as "Daddy" are talking about the 'reform candidate,' the staff members left hand changes locations on his face between shots of him front and back.
Baby Face Nelson is driving his car down the road, before approaching Everett, Pete, and Delmar. When he goes over a large bump in the road, a person can be seen in the back seat (with arms flailing) while Nelson is supposed to be the only person in the car.
While Everett eats some stolen pie, he throws his newspaper into the fire. As the first page of the newspaper burns away to reveal the second page, a piece of masking tape is visible attached to the left side of the second page. The tape holds the second page flat enough that it doesn't burn or curl quickly, so that movie viewers can read the second-page headline, "Soggy Bottom Boys A Sensation - But Who Are They?"
The guys drive to the radio station located in Tishomingo. The movie implies that it's located in the Delta country where everything else is located, but Tishomingo is actually located on the other side of the state, near the Alabama state line, over 200 miles away.
The real W. Lee "Pappy" O'Daniel had no presence in the politics of Mississippi (where, until 1975, incumbent governors were not allowed to run for reelection anyway). He was a Texas flour salesman who became a regional radio personality (as host of broadcasts of Bob Wills and the Light Crust Doughboys), then used that as a platform to launch himself into Texas politics, becoming governor, then Senator. (The filmmakers knew this anyway)
George "Baby Face" Nelson's bank robberies occurred primarily in the mid-west (Illinois, Indiana, Iowa, Minnesota, Michigan and South Dakota) with some side ventures in the San Francisco Bay area. He did not rob banks in the south.
In the movie theatre, clearly posted above each door is a modern red-lit EXIT sign, however at the time movie films were printed on Nitrate film which is highly flammable, and because of tragedies caused by the film catching fire, movie theaters were one of the first places where illuminated EXIT signs were required by law.
When the escapees suddenly realize that large numbers of Christian congregants are walking by them singing, it comes as a surprise; this could be because they walked from the church to the river, and so arrived fairly quietly.
Some viewers have noted that Ulysses is said to have seven daughters, but in the final scene only six daughters are seen crossing the railroad tracks. However, while the seventh daughter may not be present this is not a genuine goof: any number of possible explanations could be offered for the character's absence in this scene.
For the area that is going to be flooded, it is mentioned at least once that it is going to be done by blocking up a river. If this was downstream of the area it would lead to a gradual build up of water. But the flood arrives in a torrent which could only be explained by a big dam further up stream being suddenly removed. Even if the river was being redirected upstream of the area by a blockage it wouldn't explain the torrent that is seen.
There should be insulators on the transmitting tower at the radio station, but they are clearly not present on the tower shown in the film. In 1937 Mississippi, the station would be AM medium wave or shortwave, which would require insulators somewhere on the tower.
The goof items below may give away important plot points.
When the crowd is escorting Baby Face Nelson to the electric chair, two musicians are playing. One is playing the fiddle, the other a mandolin. However, the music we hear is that of a fiddle and guitar.
When the three are trapped in the barn and the police van catches fire there is a shot of a Thompson Sub-Machine Gun on top of a box, firing full-auto. This is not possible. The heat of the fire could 'cook off' the first round in the chamber. However, the Thompson is an 'open bolt' system, so the recoil of that round would drive the bolt backwards causing it to lock. Unless the trigger was pulled again a new round would not be chambered and thus could not 'cook off' again.