(At around 41 min.) Tom's S.S. tattoo is missing on the left side of his neck.
When Tom is in the car with Bud, he pats his shirt, apparently looking for a cigarette. The next shot is on Bud. When the camera is on Tom again, he has an unlit cigarette in his mouth. Tom then looks for his gun, asks Bud about it, and when the shot is again on Tom, he has no cigarette.
When Red and Julian Powell are in his fiance's house, a framed photo shows Julian and his fiance with her laughing and his head down. The first time the photo is shown Julian is on the right. The second time it's seen, the photo is reversed with Julian on the left.
All of the jellyfish at Powell's lab are kept in oblong tanks. Jellyfish cannot be kept in tanks with corners as they don't have muscles to get themselves out of the corner. The tanks should be spherical or round.
The trooper making the traffic stop at the beginning makes a serious tactical error. Police that have been taught drug interdiction, which he was clearly doing, do not search vehicles by thermselves. He clearly had backup on the way and in reality would have waited another couple of minutes before doing any type of search.