Factual error: Rob Base and DJ E-Z Rock's "It Takes Two" was not released until 1988. This movie takes place in 1987.
Plot hole: How can Bumblebee and Optimus already have the car specs and colours before they leave Cybertron for the first time to come to earth? Optimus didn't have this when they came to earth in the first Transformers movie.
Suggested correction: First of all, this might not be the same continuity as the older Transformers films. Second of all, the cars on Cybertron might be Cybertronian vehicles. Optimus scanned the semi in the first film so that he could be a different vehicle than he already was. There is no reason this could not have happened.
I'd like to add that we know for a fact that this is not the same continuity. It is a reboot. See the Wikipedia article.
Factual error: In the landing scene with Bumblebee it says it's 1987. But you see H1 Humvees pull up. Humvees were not used by the military until 1989 in Panama.
Suggested correction: HumVees were in use since 1984.
Visible crew/equipment: When the first Decepticon arrives and hits the truck and flips it, the gas cannon is visible on the underside of the cab.
Factual error: The term 'carjack' was not coined until 1991.
Factual error: "Never Gonna Give You Up" by Rick Astley is very briefly played; while that song was originally released in the UK in July 1987, it wasn't released in the US in any form, including to radio, until 1988.
Factual error: When they pull up to the cliff "It Takes Two" is playing on the radio, this song was released in 1988 from the album which was released the same year. and the film is set in 1987.
Continuity mistake: When Charlie is looking through her box of tapes, it shows the columns and there is nothing by Rick Astley. But from a different angle she picks a tape from the bottom of the middle column which we just saw, and it plays a Rick Astley song.
Factual error: In the shots of the Golden Gate Bridge with downtown San Francisco in the background, the Salesforce Tower is visible, but this building didn't even break ground until 2013.
Factual error: During the first scene with the family, dad is wearing an Izod shirt. At that time it was still Izod/Lacoste with the alligator logo.
Factual error: Shatter enables a hologram of the International Space Station from her wrist. The movie takes place in 1987 and the I.S.S. was not even started to be launched until 1998.