There Never Was Any Treasure (Situational Irony)
The biggest plot twist that takes place is after the Soggy Bottom Boys have been through their fair share of trouble. The entire plot hinges on the boys finding a buried treasure that Everett left for them somewhere near his cabin. Their search for the treasure is what has propelled the entire narrative, yet when the trio is reunited in town, Everett admits that there never was a treasure. This admission is ironic because the entire premise of the film, the men's journey to uncover some 1.2 million dollars, has been a lie, and the entire narrative is rerouted.
The Success of the Soggy Bottom Boys (Situational & Dramatic Irony)
Another irony of the film is what skilled and talented musicians Everett, Delmar, Pete, and Tommy make as a band. Without any rehearsals, they wander into a recording studio and do a perfect rendition of "Man of Constant Sorrow," that then becomes a hit on the radio. While the men do not have any history of making music together, they are somehow perfect musicians, and they make an incredible recording. This in itself is ironic, because it subverts our expectation of how a group of criminals on the run from the police might sound if they were asked to lay down a track without a rehearsal. There is comedy in what naturals the men are.
The Soggy Bottom Boys' success is also an example of dramatic irony, because as they continue on their journey, they have no idea how popular they are becoming on the radio. The viewer is privy to a scene in which a record producer comes to the recording studio wanting to know the identity of the Soggy Bottom Boys. Apparently, they've become a big hit and there's a music career looking for them. The disparity between our knowledge of their success as musicians and the fact that Everett, Delmar, and Pete are still in peril running from the law, creates a dramatic irony.
"Babyface" Nelson (Situational & Dramatic Irony)
George Nelson is one of the most notorious and dangerous criminals at large. He robs banks and murders without a second thought, but when he's called by his nickname, "Babyface," he becomes angry and insecure. He's driven into a desperate kind of loneliness that is exactly the opposite of a hardened criminal. The contrast between George's sensitivity and his seemingly sociopathic criminality is ironic, in that it subverts our expectations of what a hardened criminal would be like.
Additionally, there is dramatic irony in the fact that an audience member familiar with American history recognizes that George is a wicked and infamous criminal before the Soggy Bottom Boys realize he's a criminal at all.
Pete is Alive (Dramatic Irony)
The Mississippi heat can get to a man. As Everett rides in the back of a truck with Delmar after Pete has disappeared, he thinks he sees his friend working on a chain gang. The irony is that it is his friend, Pete. Pete has been alive this whole time and Everett makes direct eye contact with him from the back of the truck. The viewer has seen that Pete is alive, so there is a discrepancy between what Everett and Delmar know and what we know. Instead of trusting his eyes, Everett chalks the whole thing up to heat stroke.