While in Tiffany's, Andrew asks about Christmas in Ireland and Melanie says that is four months away, which would mean that it's now August or September. But for most of the rest of the film, everyone in Alabama is in long sleeves. It wouldn't be cold enough for those types of clothes.
When Melanie's parents are arguing about the volume on the television while on the phone, Pearl says, "They could have heard it all the way at Joe and Lurlynn's house," but Lurlynn is married to Clinton, not Joe.
After Melanie visits the Carmichael estate, she is walking down the street in town with the sleeves of her dress pulled down past her shoulders, not up on her shoulders as they were in the previous shots.
When Melanie is in Alabama at Jakes house you notice that the leaves on the tress are starting to change color, indicating the weather is getting colder, autumn is coming. However in cooler New York (supposedly at the same time) the trees are still completely green.
When Melanie's Father Earl returns home from the battlefield with Andrew he says, "Look what I found walking down I-58". If the movie is set in Pigeon Creek, Alabama then the only interstate anywhere close to that is I-65.
Andrew mentions to his mother that they will be staying at The Golden Cherry Motel. There really is a Golden Cherry Motel (it was seen in the movie Norma Rae) but it is located in Opelika, Alabama, which is nowhere near where this movie is supposed to have taken place.
Melanie drives past a sign on her way home in Alabama that says "Greenville". Greenville is about two or so hours from the beach. If that's near where she lives, there's no way she would be a kid and at the beach by herself or whipping by the beach after the wedding. If she lives near the beach, she wouldn't be driving through Greenville.
When lightning strikes sand, or sandy soil, it fuses the sand into a glass-like solid mass known as a "fulgurite." Real fulgurites are usually partially hollow tube-like structures having a granular exterior. The clear glass objects shown in this film are not likely to have been created by lightning.
Melanie's outfit changes in the fair scene from a dress with cowboy boots, to blue jeans and a jean jacket. Update: Several hours had also passed by with it being light out when she was wearing the dress to being late at night when she was in her jeans. It's a small town so it is possible that she went back home, changed clothes and came back to the fair.
Earl (Melanie's Father) refers to I-58. There is in fact no highway numbered 58 in the United States Interstate System. There is, in fact an I-58 that spans the width of Virginia from coast to westernmost tip. But not Alabama as is referenced in the movie.
Jake bumps a gravestone which wiggles back and forth, showing that it is made of a light material such as foam board (apparently it does not wobble as though it were a loosely fixed stone as some have suggested).
In the graveyard, after Jake tells Melanie to leave, she bumps into the "stone" dog on top of a grave marker, which wobbles more than a loosely-fixed decoration would, revealing that it is made of foam.
The goof item below may give away important plot points.
The big revelation of this movie is that Melanie has been living under the surname "Carmichael" in NY, and that is the name the fashion industry knows her under. But several of her former friends in Alabama reveal that they know of her designing career; one of whom even says he's read stories about her on the Internet! Why would the town be so so surprised then to learn that she had been living as a Carmichael instead of a Smooter?