Factual error: During the scene where everyone runs outside, the Statue of Liberty head crashes into the street. The head is scaled too small and its size can be compared to the people taking pictures. In real life the statue head is large enough for people to walk into (17' from chin to cranium).
Audio problem: When the Statue of Liberty's head strikes the top of a distant building before landing in the street, sight and sound are simultaneous. There should have been an audible lag in the sound due to the speed of sound being only about 1000 feet per second. The absence of this lag would be acceptable in most movies but is inconsistent with the amateur home movie theme of Cloverfield.
Continuity mistake: In the scene where Hud, Jason, and Rob are hiding in the grocery store, you can see several items toppled over on the shelves above Jason's head. When the lights start to flicker, you notice almost all the objects are now standing upright, only to be toppled over again.
Continuity mistake: In the scene where Rob is walking on the street in the direction of Beth's apartment (just after leaving the electronics store), you see on the right-hand side of the street the orange framework of a store front. There are parked cars there, including a black Chevy Malibu followed by a white Mercedes, which Rob stands in front of as he points towards the military on the next block. Two scenes later they are (briefly) walking the opposite direction on the same street, which would be away from Beth's apartment. The orange framework is now on the left side and the direction of the parked vehicles is reversed. The black Malibu and white Mercedes are in different positions (as are the other cars on the street). I don't know New York City, so I don't know if the buildings in the background are different. Also, a man in a yellow shirt and tan pants runs past them in both scenes. It is the same man both times.
Factual error: In the first scene where the Army attacks the monster, the first tank that rolls by is actually a British FV-433 Abbot self-propelled gun made during the Vietnam war years, not an M109 Paladin which most people believe. (Paladin has a longer gun barrel, and has rear turret overhang.) And a few seconds later we see, next to the M1 Abrams, what appears to be an MLRS system firing, but is actually some kind of truck with a multiple rocket launcher, much like the Russian BM21, not the U.S. tracked MLRS system or the M1097 Avenger (which I originally thought it was).