In the opening scene, the plane externally is a Boeing 737-700 and when the camera zooms into the windows looking in, you can see that the seating arrangement is 3 seats an aisle and 3 more seats - the normal configuration for this aircraft. However when the camera zooms inside the aircraft, the cabin layout is much bigger with a 3-3-3 seat layout and dual aisles. The only twin engine aircraft that Boeing has that could match this seat configuration is the Boeing 777. At the end of the scene just as the plane is landing in the sunset, a completely different aircraft - a Boeing 767 is shown landing.
When Sergei dies, he drops the microwave gun right underneath the alien that 'dusted' him, however, just a moment later, Ben picks it up as it is now laying at his own feet and now has a blue, a red, a green and a yellow light on it. Towards the end, just before jumping into the river, the original microwave gun with the timer dial on the side returns.
The radio that the group finds in the US embassy works because it was protected from the EMP strike inside the bird cage which acted like a Faraday cage. A Faraday cage blocks EM waves, including microwaves. The radio works in the bird cage because the antenna is connected to a wire leading outside of the cage; but it would not have worked in Sergei's apartment.
When the electric powered bus, the 'trolleybus' breaks free with Sean and Natalie on board, the bus makes obvious diesel engine sound, although these do not have backup engines, and the logic of the scene is, that the bus moves without the overhead electricity line feed because the alien supplies power.
In the mall, Sean discovers that the aliens cant see them through glass since they are reading the human bio-electrical pulses and glass does not conduct electricity. So the bulb on Sean's neck shouldn't be lighting up either.
The Moskva and Oka Rivers which traverse the city of Moscow,are not deep enough to be navigable by ocean-going vessels. A nuclear submarine simply could not travel down it as far as Moscow without running aground.
Toward the end of the film when the characters have to jump off the boat into the river, Sean and Natalie both jump in together holding hands. All the characters except Natalie emerge together and climb aboard the submarine. Somehow Natalie has managed to end up in the city, clearly more than a few kilometers away.
If Sean and Ben were planning on selling their software to Russian investors where were their attorneys? On either end..the US or Moscow?
Even assuming that the film was accurate and they were both woefully naive and inexperienced, how were they going to sign a deal without having attorneys prepare the documentation or review what was signed?
Sergei has transformed his apartment into a giant Faraday cage he built as a protection against the aliens. It is not likely that he and Vika (if she contributed at all to this) could have both set up the cage during the four first days of the invasion, considering the huge amount of material needed.
When they are floating down the river on the boat to get to the submarine, you can clearly see the footage is reversed to make it look like the boat is floating down the stream. The sailing route is clearly visible, and the boat is "floating" backwards along it.