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 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.
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.
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.
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.
When escaping from the burning barn with the Hogwallop boy, he is shown to drive an early 1920's Ford Model T. While the Ford Model T has three pedals and looks similar to a modern car, it is operated in a completely different fashion. The boy is shown to step on the gas, but in a real Ford Model T the throttle is located on the steering wheel while the pedals operate the brakes and first/second gear.
Flash flooding only occurs when a dam breaks, not when it closes initially.
Several times the characters refer to the amphibian that Pete was changed into as a toad or horny toad. In every shot it is clearly a frog, not a toad. Near the end of the film they did get it right and correctly called it a frog.
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.
When the they arrive at the radio station WEZY the aerial shot shows no power lines anywhere along the roads or coming across the farm fields to the building. It is unlikely there was an underground line powering the building. Even with no obvious source of electricity all the lights and recording equipment are electric.
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.
During the picnic, you can see the false branch that Big Dan is about to rip off the tree.
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.
All of the records shown in the film are turning at 33-1/3 RPM. In 1934, only 78 RPM was used. The first 33-1/3 RPM album was released in 1947.
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)
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 recording of "Man of Costant Sorrow" Ulysses steps away from the microphone to let Pete and Delmar sing their part but we hear three part harmony.
While sitting around the campfire, Tommy Johnson is playing his guitar. The sound of a resonator guitar is heard but Tommy plays a standard guitar with a sound hole rather than a resonator top.
During the KKK meeting, when Ulysses and Delmar are following Tommy to the noose, the clan members are marching on the spot, but the sounds of marching do not match the movement of their feet.
Despite Chris Thomas King being an actual blues guitarist, the scenes in which he is accompanying the Soggy Bottom Boys, his guitar playing is very obviously not in sync with the sound track.
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 camera is reflected in a window of the locomotive that passes in front of the three escaping convicts.
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.
Menelaus "Pappy" O'Daniel mentions, "parishes and precincts". Louisiana is the only state that uses parishes, the movie is set in Mississippi.
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.
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.
At the KKK meeting when Goodman catches the flag, the leader says don't let that flag touch the ground. But after Goodman catches the flag he sets the end with the flag on it on the ground.
Nelson asks if any of "you fellers" knows his way around a Walther PPK. He then pulls out a sub-machine gun. The Walther PPK is a pistol.
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.
In the Klan rally scene, shortly after Big Dan catches the flag pole to prevent the confederate flag from touching the ground, he lowers the flag pole with the top pointed down, so that the flag would clearly be touching the ground.