Plot hole: The huge explosion that rips through Tara's house isn't very likely. She only left the stove on for a short while, and if there was really a massive concentration of gas as depicted by the explosion, the Agent who fired the shot that triggered it should have collapsed from inhaling so much gas.
Continuity mistake: When Graeme gets shot towards the end of the movie, there is a large bullet hole, with visible blood, and a torn jacket where the bullet went in. Later when Paul is getting onto the space craft waving goodbye, the bullet hole, blood, and suit are now all gone and cleaned up. As if it was never there. Taking into effect of what we know, Paul can heal, but I don't think he can mend clothing as well.
Other mistake: When Graeme is convincing Ruth to go with them and she says "Where would I sleep?", you can see her left eyeball through the dark lens: it moves as she looks to her right twice and then back. In the next scene, it is shown that she doesn't have a visible eyeball in her left eye.
Continuity mistake: As Nick Frost and Simon Pegg are running away form the fireworks shooter, they both have their head lights on. As they reach the others Nick removes his and in the next shot he removes it again. Simon has his on in the first shot, it has then disappeared in the next.
Continuity mistake: In the scene at Ale inn where they are deciding on which banner, there is a gentleman sitting at a table in the background. He is served by a waitress, there may be another person there. He and the waitress chat while it looks like she is pouring coffee. Next cut to Simon and Nick shows that the waitress and customer have vanished.
Factual error: When a fight breaks out in the Wyoming restaurant, there are a number of sailors shown in summer whites. There are no naval bases in that state, and military personnel on leave would not be wearing their uniforms.
Continuity mistake: After Graeme and Clive leave the diner, Graeme gets in the RV after he said he was putting the alien on board sticker on the back. But after they crash into the pickup truck and drive past the ET highway sign, the sticker can't be seen on the back of the RV. Neither can the black mark that is left behind when Graeme later removes the sticker after they meet Paul.
Continuity mistake: In the scene in the RV where Paul and Ruth are arguing about religion there is a fly swat and a green flowery oven mitt hanging on the hooks on the wall behind Ruth. In the next shot, she has the fly swat in her hand but the green mitt has disappeared, and it isn't she just knocked it down out of shot, because later in the same scene when we see Clive driving and he says "Oh yeah" it can be seen hanging there again.