When Max is in the bar drinking a shot and a beer and meets Bree for the first time, as he gets up to leave the glass with the whiskey has about an inch of whiskey in it. Max leaves and the camera cuts back to the bar and the whiskey glass is now empty.
In the beginning of the movie, Will Caster is putting up a copper-netting to create a zone where there's no wireless signals. This is called a Faraday Cage and would not work as depicted - it needs to be a complete cage of thin copper-mesh, not just a web that's hanging from some poles.
The doctor at the hospital says that the bullet which hit Will contained "an isotope called Polonium". Actually, Polonium is a chemical element, while an isotope is an atom with the same number of protons of a chemical element, but a different number of neutrons.
When Will Caster is leaving the hospital and brought out in a wheelchair, he is wearing a hospital scrub for his shirt. This is highly unlikely because hospitals issue examination robes, not doctor's scrubs, to patients. Hospitals guard their supply of scrubs and keep them in doctor's changing rooms, not in the ER or patient rooms.
Will Caster is killed by the highly toxic radio-active element, Polonium (the same one used to murder Alexander Litvinenko in true-life in 2006). Despite its toxicity (scientists estimate that 1 gram could kill 50 million people), his wife and friends are allowed to remain with him in close proximity until his death. Whilst intact skin is actually a barrier to the passage of alpha radiation particles to a nearby person (so we could let them off this goof), we later see Caster's cremated ashes being tossed into the breeze above a river for all to breath in. These are hardly actions that any homeland security or radiation expert would conceivably have allowed to happen.
A Turing-complete A.I. with the capabilities of that depicted within this movie would require storage space of the magnitude of Petabytes, if not Exabytes. When Evelyn hastily uploads the "Will" consciousness to the Internet, the process takes only a matter of a few seconds (and this over a satellite Internet connection.) To actually upload such an amount of data using the best (as of 2016) fiber-optic Internet connection would in fact take years, if not decades.
The goof item below may give away important plot points.
Poisoning by the radioactive element Polonium would have caused Will Caster's hair to completely fall out within a few days at most. Yet even when he's ready for the upload procedure It's clear he still has a healthy head of hair which has had to be shaved.