Continuity mistake: When Juliette Binoche is running for her life in the tunnel near the beginning of the movie, when the running starts you can see all men run pass Juliette and then she's the last runner. There's even a shot when she's looking back at the leaking water and is last, with no-one behind her. A second later she's in the front of the running group.
Factual error: San Francisco Bay's deepest spot is 360 feet deep and that is directly under the Golden Gate bridge. Godzilla is only 355 feet. His head would be under water when he stood up in next to the bridge. Also the bay is fairly shallow everywhere else - most parts only reach 45 to 50 feet deep. His dorsal fins alone at 34 to 37 feet tall. So there is no way he swam all the way up to the bridge and then stood up in the water with it only being below his belly. He also couldn't have displaced that much water for it to cover him.
Character mistake: All the characters wearing gas masks are wearing them with the straps over the hoods. I was army trained, and you always wear the mask next to the skin for best seal, with the hood over the top.
Factual error: During Ford's rescue from the boat Search and Rescue is able to find him, hoist him, and vacate the area within the 5min left on the nuke. Having done this in real life it takes a tad longer.
Factual error: When the nuclear bomb goes off, the main character is on the rescue helicopter in clear view of the blast. They are close enough to the blast, with the view they showed, that the helicopter would have been blown out of the sky.
Factual error: When Ford and his dad are in the Blackhawk/Seahawk, they are talking at just above a whisper. There is no way to "whisper" in a black/seahawk.
Factual error: In the scene where the SEALs perform a HALO jump from what appears to be 27,000+ to 30K feet, (based on their average on-screen time in freefall), they are not wearing balaclavas over their heads, thus exposing their foreheads and necks to the sub-zero temperatures at that approximate altitude, added to the significant wind-chill effect of 110 to 127 MPH in flared freefall at the thin atmosphere of those heights. Frosbite would have been almost instantaneous.
Other mistake: When the father and son sneak into the quarantine area, they are wearing radiation masks. The father's mask is a slip over face mask (not entire face, hood and shoulders one-piece like the son's) that shows some exposed skin in a gap between the hood and mask, yet they speak about how severe the radiation is expected to be. The father using a meter detects no radiation and he dramatically slips that mask type off to breath the air. If there was severe radiation as they anticipated, he would have already been harshly exposed.
Factual error: In the scenes on the railroad bridge: (1) the gap between railroad ties varies from 18 inches to 6 inches. Ties on a railroad bridge usually have an 8 inch gap. (2) The muto is stated to be 300 feet tall. In different scenes the muto appears to be taller, shorter and the same height as the railroad bridge. (3) A diesel locomotive averages 60 feet tall which is 1/5 the height of the muto. When the locomotive careens over the end of the bridge, it appears to be much smaller. (4) Railroad flat cars average 60 feet in length making the incomplete nuclear missile on the car almost 50 feet. They are not this large. (5) The railroad bridge appears to be spanning a river but its design is terrible. It would fall down. In the scene with the beached russian sub: the sub is probably a typhoon class which is 500 feet long and 75 feet wide yet the muto towers over it.