When Nick is talking to Dr. Elsie Chapman during the jeep ride in Panama, it is obvious that the whole scene was filmed on the same short stretch of road; the same characteristically tall, slender tree can be seen behind both actors several times, in both the shots and reverse shots.
The very first footprint (the one that Nick walks into) is clearly from a different animal than the ones that followed. The footprint is rather solid pad-like with protruding toe-like appendages whilst the ones that followed it are more talon-based phalanges or fingers. Much like the differences from an elephant's foot and a bird's. This is also the only time this footprint takes place.
Elsie suggests that Godzilla's footprints may belong to the dinosaur Allosaurus, and Nick corrects her that Godzilla is much too big to be a dinosaur. A true paleontologists wouldn't have considered this in the first place, as Godzilla's footprints are already larger than most dinosaurs. It's also impossible to determine the genus of a dinosaur from just a footprint, so Elsie's guess couldn't have been any more specific than a "theropod dinosaur" (bipedal carnivorous dinosaurs).
Throughout the chase scene near the movie's end, the advertisement light on the taxi roof is on/off between shots. In addition, sometimes it appears firmly fixed in place, while at other times it is obviously loose and sliding all over the roof.
When the Broderick character leaves town in a cab we see pouring rain through the plain-glass rear window of the cab. Later, when the cab stops to reveal the driver as French secret service, we see the rear window over Broderick's shoulder suddenly now has defroster wires embedded in the glass.
When Dr. Niko is standing in his "sample" (i.e., the footprint), Colonel Hicks uses a ladder to climb out of the footprint, but they never descended a ladder to get into the footprint in the first place. A long shot of the scene reveals that all sides of the print are vertical - there is no sloped side that they could have walked down.
Upon showing the aftermath of Godzilla's first attack on New York there's a shot of the MetLife building with a huge hole torn right through the middle of it. But later there's a wide shot of the city showing the MetLife building completely intact.
The cargo ship seen washed up on the shore after Tatopoulos examines the footprint is different to the ship attacked at the start of the film. The ship at the start has a single superstructure at its rear, whereas the washed up ship has front and rear superstructures.
After Madison Square, characters drive toward the Brooklyn Bridge because it's "the closest suspension bridge" to their location. In fact, they'd have to pass the Williamsburg and Manhattan Bridges - both suspension bridges - to get to the Brooklyn Bridge.
The missiles used do not have enough explosive power to blow the top off the Chrysler building. Additionally, the missiles do not have the power to destroy Madison Square Garden in such a huge explosion with just two shots.
The AH-64 Apache helicopter does not have a machine gun mounted on its "chin." The gun it carries is located underneath the fuselage. Also, the muzzle flash from the guns mounted on the side would interfere with the pilot and the gunner's night vision.
When fighters are locking onto Madison Square Gardens with their missiles, they say "Selecting LGB's" (Laser Guided Bombs). On the screen the pilot is looking at it says "Weapon AGM-84 HARPOON (2)." Harpoon is a radar-guided anti-ship missile, not a laser-guided bomb. Later the F/A 18s are commanded to release their Mavericks, yet another type of missile - primarily an anti-tank weapon.
In the cockpit, we see that the machine guns fire in pairs; Apaches only have one automatic cannon. Also, from an outside view, we can see the fired rounds coming from the wings; the Apache's gun is mounted underneath the nose.
Jet aircraft cannot fly so close behind each other when in a line formation; the engine exhaust from the first Jet would extinguish the seconds jet engine, having momentarily eaten up all the oxygen in that path.
"Cold blooded animal" does not mean its blood is cold; it means it does not regulate temperature by itself. To move a mass of that size, you would need a lot of energy and you probably would dissipate a lot of heat (this goes for the tracking missiles, and the fact that the animal seems to move very fast in cold water). Also, it would probably be easy to track on land with seismic triangulation.
The military helicopter that picks up Tatopoulos from Chernobyl at the beginning of the movie is an American built Piasecki H-21, known also as "flying banana." However, neither Ukraine nor the former USSR ever possessed or used this type of helicopter.
The officers aboard one of the submarines wear caps reading "USS Utah SSBN 745". "SSBN" refers to ballistic missile submarines, which are large, heavy and slow, therefore unsuitable for pursuit tasks. A fast attack submarine (SSN) would make more sense.
The opening sequence shows some mushroom clouds and captions telling the viewer they're looking at French Polynesia. The French nuclear program existed only of subsurface explosions. The explosions shown are actually American nuclear tests from the fifties.
At the end of the movie when Godzilla is chasing the cab across the Brooklyn Bridge, the bridge only has 2 suspension cables (1 on each side) whereas the real bridge has 4 (1 on each side and 2 running down the center).
After Godzilla jumps in the water, in the first scene when the camera passes by the first submarine, an American flag is clearly visible painted reversed onto its hull. No submarine in the US Navy, or any other navy, have flags painted on them.
The goof items below may give away important plot points.
When Nick buys the home pregnancy test, one of the boxes given to him by the pharmacist is a "Duane Reade" brand box. Dunae Reade is a large chain of drugstores in NY that sells their own brands. Small drugstores do not sell the Duane Reade brand.
Most of the last hour of the movie is concerned with trying to keep the hatchlings contained inside Madison Square Garden, but they ignore the fact that there's a huge hole in the floor of the arena leading to the sewers. The search team has to know about the hole - it's how they got into the building themselves.