Graham asks Bo to turn up the volume on the TV, she reaches forward and the volume increases before she makes contact with the button. Also, she then holds her finger on the button but the volume does not continue to increase, instead, it remains constant.
When the police woman gets into her police car after telling Graham to take the family to town, the car engine can be heard starting up early, before she is fully in the vehicle, yet she didn't appear to reach out to the ignition.
When Merrill yells at the children on TV at the birthday party in Brazil, he uses Spanish - "vámonos!" ("Let's go!"). But people in Brazil speak Portuguese, so the correct term would be "vamos!" However, it is a common misconception that Brazilians speak Spanish and could simply be an error on the part of the character.
While sheltering in the basement of the house belonging to Mel Gibson's character, he and Joaquin Phoenix's character are looking for a coal chute - something Gibson's character would have already known about, having lived in the house for several years.
The basement door is green in every shot except the very last one when Merrill is opening the door to go up and check for the "all clear" - it is a close-up of the handle and the door is painted brown.
When they drive into town, we are shown an aerial view of East Lancaster Avenue and the Oxford Presbyterian Church in Oxford, PA (near Maryland), when the area the movie is supposed to take place in is Newtown, PA (near Trenton, NJ).
There is no such agency as the Bucks County Police. All towns in Bucks County, and almost all of Pennsylvania for that matter, are served by either local police, regional police, or the Pennsylvania State Police, never by county police or sheriff's offices (which are restricted to court-related duties). The only county police force in Pennsylvania, the Allegheny County PD, is on the other side of the state from Doylestown, and furthermore, is restricted to providing law enforcement services only on county property.
Officer Paski's uniform features an American flag on the right arm which is backwards. Reverse flags are to be worn on the right arm while the left arm uses the standard flag direction. The stars of the flag are always supposed to be closest to the heart as a rule of thumb. On Officer Paski's right shoulder the flag is incorrect and a huge mistake.
They nail boards on the outside of inward-opening doors. Obviously this isn't going to prevent the doors from being opened, but anything is better than nothing and they're panicking anyway. Moreover, when Merrill asks Graham how they will know if boarding the windows will work, Graham replies, "Because they seem to have trouble with pantry doors". Graham is implying that if the aliens have trouble with pantry doors, boarding them can only help. Doing so also may give everyone, especially the children, a sense of hope and/or security.
Many viewers will ask why aliens who are harmed by water would invade a planet, unprotected, that is 70% water and has rain. It's likely that we don't know what their situation is. If they are desperate for resources, they may take a calculated risk. We already know their crop circle map was nowhere near water. It's also unlikely that they would know we would figure that out and use it against them. The fact they invaded all at once shows they were not planning on a prolonged engagement, but more of a blitz attack and would not have known if we would fight back or if we would even figure out something as simple as water would be an effective weapon.
The argument about aliens capable of flying millions of light years but not being able to open doors is not valid. At no point do we know they can't open them. The alien in the pantry isn't necessarily stuck because it can't open the door, it could just be waiting to make sure it's safe to leave. This is evidenced by the fact the aliens turned the knob on the basement door immediately. Graham's opinion that they have trouble with pantry doors is just that, an opinion and a guess. The alien in the pantry clearly got out because that's the alien Merrill beats with his bat.
When the Hess family goes into town, most of the trees in town have changed into their fall colors. Indeed, some leaves are brown, and there are dead leaves in the street. Yet, most of the trees around the farmhouse, especially those surrounding the corn field are dark green.
When Graham drops the knife in Ray Reddy's house (when the alien lunges at him) the hand that fumbles for the knife wears a skin tight, dark colored sleeve. Graham's jacket in the scene is baggy-sleeved and light colored.
The goof items below may give away important plot points.
Boom mic visible
As Graham takes Bo inside, after Houdini has died, they meet Merrill on the way into the house. As Bo mutters "Houdini's sick", the shadow of a boom mic can be seen traveling across Merrill's shirt and face.
When Graham comes to the children after Morgan had to kill their dog, Bo is sitting with her knees up to her face. In this shot her shoes are perfectly tied. Once it changes to Graham and Morgan looking down at the dead dog and Morgan leaves, Graham goes over to Bo. Her right shoe lace is untied.
At the end of the movie when Merrill is swinging the bat, the bat breaks. It shows the break being a somewhat clean break in the middle of the bat. Baseball bats do not break that way since they are made with the grain of the wood. This causes them to splinter when they break, and they break at the weakest part of the bat - the handle.
Aliens are supposedly afraid of or damaged by water, yet they can also run naked through the best way for a body to become soaked: a Pennsylvania cornfield at night. It would be highly, highly unlikely (if not unheard of) that there would be no dew on the corn at night.