When Dodge and Penny go to Olivia's parents' house, Penny breaks the window. They then enter the house, and you can see the window in the background, unbroken. Then a few seconds later, another shot occurs and the window is broken again.
Dodge's cleaning lady is at his home on the day the asteroid is going to hit. She says it's a Thursday, and yet the news announcer tells everyone to set their clocks forward that night for Daylight Saving Time - which always happens on a Saturday night/Sunday morning.
In the last scene of the movie where Dodge and Penny are in bed. You hear the sound of the meteor impact, which after a minute or so has the shock wave hit. If the meteor is as large as they say, the shock wave would arrive first obliterating the house before the sound would have ever reached them.
Speck has several Smart Cars for his fleet for use after the post-apocalyptic world. However, these vehicles are parked above ground, in the open, outside his shelter and Speck should know that all these vehicles would be incinerated by the effects of the meteor.
Dodge pulls out the vinyl album "Scott", Scott Walker's first solo album. He takes record out of sleeve puts it on turntable but "The Sun Ain't Gonna Shine Anymore" plays. This was by The Walker Brothers and was not on the album in question. Additionally, the following song that plays "Stay With Me Baby" does not appear on the album "Scott." Nor does it appear on the album "The Sun Ain't Gonna Shine Anymore." The two tracks only appear together on later released compilation albums.
Coincidentally though, the first track on the album "Scott" (the one shown but not played) is "Mathilde" - an alternate spelling of Matilda - which is the name of the asteroid in the movie.
The goof item below may give away important plot points.
The plane Dodge loads a sleeping Penny into is a Cessna 2-seater (either a 152 or a 150). Both models have a max range of around 400 miles. Neither would make it to England from anywhere on the east coast of the US.
The announcer on the TV at the end says that the asteroid Matilda is arriving one week earlier to Earth than expected. If this were the case, then due to the trajectory of an asteroid and the velocity at which the planet moves in orbit, the two would no longer collide with that great of a mistake in timing unless either there was a mass conspiracy to lie about the timing of the asteroid or for some reason the planet stopped moving in orbit for some undetermined reason.