John Kramer states that Anna framed her husband for the death of their baby. But John cannot know this to be true. He overheard the couple arguing, but there is no possible way for him to know that Anna's husband did not roll over and suffocate their baby. Similarly, John claims that Ryan lied to the police regarding the car crash that killed his best friend. Unless John was a first-hand witness to this event, he could not possibly know the truth.
When Logan mentions that hydrofluoric acid was used to kill the second victim, he mentions that the tissue is connected to the heart, therefore everything, including the heart, is eaten away by the hydrofluoric acid. This is incorrect as hydrofluoric acid only eats oxides, and not organic tissue like flesh. If dropped on the skin, nothing would happen. It would seep into the body and begin dissolving the bones of the victim by a process known as decalcification.
Since most of Jigsaw's voice is pre-recorded on tape and on TV, it would be hard to know the victim's name if they accidentally fall into the wrong trap. A little while after Ryan get his leg stuck in the Leg Wires underneath the floorboard the TV's come on (not his trap or test). Jigsaw mentions Ryan's name. Since the Leg Wire trap was made for anyone who tried to go through the exit, he would have no idea who's name it was.
Detective Halloran looks up Logan on the Department of Veteran Affairs website and is able to pull up Logan's medical records, it lists his diagnosis and prescription history. This is all information that is protected by doctor/patient privilege, it wouldn't be posted online for anyone to see. Even if Detective Halloran was accessing a database only available to law enforcement he still would not be able to access Dr. Logan Nelson's medical history without Logan's written permission or a court order.
After the group of people make it through the Bucket Head Room (buckets on heads and circular saws on the door); the main part of the Jill Tuck's barn looks like it's almost entirely made of wood. If the walls were made of wood, the characters could have found a way out considering that there are enough tools around.
The goof items below may give away important plot points.
While Detective Halloran is wearing the Laser Collar trap, it is shown to be equipped with seven lasers (there is a point of view shot from inside the trap). However after the trap kills him, his head splits into eight sections instead of seven.
When the police go to dig up John Kramer's grave, they should have noticed that it had been dug up within the last few days, since Logan would have needed to dig up the coffin to remove John's body and replace it with Edgar Munsen's.