When Lily is pretending to be Hannah in Miley's bedroom, Miley says Hannah can only fly west to east, so she has to take the long way. Flying directly from Los Angeles to Tennessee IS flying west to east.
Miley gets her luggage out of the truck with her beauty kit is in her left and her dress is in her right. Miley stops and faces her father. Then her luggage was inverted. But if you look and observe closely, the camera turns with Miley and the luggage is not moved.
When Bobby Ray is collecting the dishes at Grandma's party, he has 2 handfuls of dirty dishes with silverware and napkins on each one of them. But when he shakes Lorelai's hand he only has 4-5 plates, which appear to be clean and without the napkins and silverware.
Immediately after the first verse of song Hoedown Throwdown, Miley picks up the mic from the stand with her right hand, then within two seconds the mic is placed on the stand and not in her right hand.
When Miley is changing out of Hannah Montana next to Blue Jeans, her horse, it shows her taking off everything that would link her to Hannah Montana. After she closes her suit case she still has her big yellow earrings in but the next shot she doesn't have them in.
In the scene where Miley ran to the chicken coop before Travis could find her and pretended to be painting, she bent down to put paint on her pants, and you can see that there is already paint on her shoulders of the blue sweater just before she brushed them with the paintbrush. In the previous scene where she was wearing that sweater, there wasn't any paint on the shoulders.
The town in Tennessee is supposed to be a quiet little farm town. They filmed the main street scene in Columbia, TN, a city of over 33,000 people. If you look carefully in the background you can see a busy highway or boulevard most likely used by commuters going to Nashville.
Near the beginning when Miley is leaving gym class and stops at her locker you can see her pull it open a little and the fiddles with the lock, as it unlocking it. But if it was really locked to begin with it wouldn't have opened at all.