Heavy kudzu growth is visible in many scenes - this is an invasive plant species introduced to the United States in the early 20th century, and it's much more common in Georgia where the film was shot, than Appalachian Virginia where the film was set. And it probably wouldn't have been in either of those areas at all in the 1930's.
The movie is set in the 1930s, at one point a character is loading a shotgun with plastic shotgun shells, but plastic shells weren't introduced until the 1960s. During the period of the movie shotgun shells were made of thick paper with a metal bottom for the primer.
Telephone (utility) poles shown in several scenes include both telephone lines and cable TV lines, some of which include modern "expansion loops." Cable television wasn't introduced in the US until 1948.
Early Galvin radio in dash of the '32 Ford V-8 was not correct because at that time radios had a separate control head with the radio mounted under the dash. Also, the horizontal dial was not accurate to this time.
Jack leaves his left shoe at the church and while walking to see Cricket his missing shoe is plainly visible. When Jack is assaulted by Special Deputy Rakes he is shown with both shoes on even though he doesn't get the shoe back until much later at Black Water station.
When Floyd Banner is introduced, he shoots down several men in a car in the center of town with a Thompson submachine gun. Near the end of the scene the rear of the car is shown with bullet holes in it, but Floyd Banner was never behind the car, and he could only have put holes in the side and front of the car.
The flowers at Cricket's funeral (ranunculus, wax flower, French anemones) would likely not have been available in the rural mountains of Virginia at the time. They would have likely used wildflowers such as goldenrod and queen anne's lace.
Special Deputy Rake assaults Jack and repeatedly punches him on the left side of the face however later on for the rest of the time that the bruises are visible they are located on the right side of his face.
The goof item below may give away important plot points.
During the shootout at the bridge, the sheriff shoots Agent Rakes, seemingly in the back of his left leg. But when Rakes retreats to the bridge, there is no sign of a bullet-hole or blood anywhere on the back of his suit.