In both scenes with Brian, Aron's boss, he is wearing a T-shirt that can be identified as being a product of the website www.someecards.com. This website did not launch until 2007, but the movie is set in 2003.
In Aron's guidebook, at the top left of the map, there is an inset with an elevation profile. On the elevation profile, "Aron Ralston's chokestone" is labeled. This label actually appears in current versions of the guidebook.
(at around 1h 23 mins) When Aron plops into the water pool after freeing himself, his hat comes off in the water. Then he rises to kneeling, drinking and splashing water on face and head. In at least one quick shot of this, his hat is on.
After crashing with his bike in the beginning of the movie, Aron clearly lies a couple of feet away from it. When he takes a photo of himself a few seconds later, however, he suddenly lies with his head on top of the bike.
When Aron and the girls are taking a the photo, feathers are visible in the girl's hair. When he snaps the photo, the feather has jumped to the other side of the girl in green's head. Her necklace is outside of her shirt then tucked in, and the rock in the background is differently colored and patterned.
When Aron is watching the film of Kristi and Megan in the pool on his camera, it moves too far to the left, so a hand is seen pushing the camera back into shot. This can't be Aron's hand as he's holding the camera.
After the boulder initially lands on Aron's arm he decides to remove his backpack to see what he can use to help him get out of this situation. He pulls off the backpack from both shoulders with a fair amount of ease despite his right arm being trapped under a rock and incapacitated. However, although not shown, all he had to do was undo the buckle on the right shoulder strap.
In the initial scene where Aron tries to move the boulder it can be seen to wobble slightly. However, it's described in his book Between a Rock and a Hard Place that when Aron tries to first move the boulder it does wobble slightly, trapping his hand further. He then proceeds to move the boulder back to release the added pressure he caused by moving it.
When Aron rides his bike off the top of his truck, the camera pushes through the back doors of the vehicle. This vehicle appears to be a cargo van (similar to one that crew may use for transport) with doors that open from the center. This cannot be Aron's vehicle as he drives a truck with tailgate and a canopy.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
The "127 hours" of the title refers to the time of his uninterrupted experience, beginning with when he woke up that first morning in his truck until he passed out from the morphine in the hospital after being rescued, not just the time trapped by the boulder.