While Charlie's mom is driving Bumblebee, she freaks out when Charlie pulls up next to the driver's window in her moped and says that she was afraid she was being carjacked. The term carjacking was not used in 1987. Per Wikipedia: The word is a portmanteau of car and hijacking. The term was coined by EJ Mitchell, an editor with The Detroit News. The News first used the term in a 1991 report on the murder of Ruth Wahl, a 22-year-old Detroit drugstore cashier who was killed when she would not surrender her Suzuki Sidekick.
In wide shots of the Golden Gate Bridge with downtown San Francisco in the background, the skyline-dominating Salesforce Tower can clearly be seen. This building did not break ground until 2013, over twenty years after this movie takes place.
When testing Bee's new radio/tape deck, Charlie briefly plays a tape of "Never Gonna Give You Up"; although the song is from 1987, the year of the film's setting, it did not release in the United States until early 1988.
In one sequence, Bumblebee is seen crushing a can of Tab Cola with his fingers. The can however is of modern vintage, as shown by the can's shape (with the tapers, as opposed to the more purely cylindrical can of 1987) and the calorie count mark on the label (which was added sometime after 2010)
When Charlie and Memo go to the big cliff-side party, at one point Memo says the phrase "What's your damage?" to one of the bully girls. This phrase was popularized by the movie Heathers, which came out in 1989, 2 years after Bumblebee is set.
Dr. Powell remarks that the Decepticon modifications to the Sector Seven computers have created an "interconnected web of information," implying that the technology behind the Internet originated from the Decepticons. However, the idea of an interconnected network of computers for communications and data transfer had been developing for decades prior to the film's setting of 1987, and the term "Internet" first appeared in the mid-1970s.
During the mid-credits scene, Bumblebee meets with Optimus Prime and they witness other Autobots arriving on Earth. In the original film, none of the other Autobots or Prime himself arrive until 2007, two decades after the events of this film. Plus there are more Autobots coming down than in that film.
When Charlie first turns on Bumblebee in its WW Beetle form at her uncle's scrapyard, she's wearing shorts. Then she's seen driving it home from there, but when she exits the car in her garage, she's wearing ankle length jeans.
Near the end of the film, Bumblebee is shown driving on the Golden Gate Bridge alongside Optimus Prime, appearing as a 1977 Frightliner. However, the first film establishes that Prime did not arrive on Earth until 20 years later, in 2007, choosing a 1994 long nose Peterbilt as his form.
It is implied that the fictional town in which Charlie lives is located north of San Francisco, as several locations in the northern part of the San Francisco Bay Area are seen throughout the film, including San Pablo Bay, Point San Pedro, Rodeo Beach, and the Golden Gate Bridge. It is highly unlikely that she would drive a moped to work at the Santa Cruz Beach Boardwalk, over 90 miles to the south.
In a scene where Charlie is introducing Bumblebee to different music, she opens her cassette tape holder and there are numerous cassettes by her favorite band, The Smiths. One of the tapes seen there is "Rank". However, this particular live album wouldn't be released until September 5, 1988 - a year later. Which means, this would never have been in her collection back in 1987.
Charlie starts playing the first track (of a vinyl record) on Sam Cooke's album Sam's Songs, which ought to have been "Little Things You Do" rather than "Unchained Melody". "Unchained Melody" (the song heard in the film) doesn't even appear on the album, being the last track on side A of Cooke's earlier release "Hits of the '50s."
In the original movie continuity, Bumblebee is said to have had his voice damaged in a battle with Megatron and Ratchet had been trying to fix it. In this movie his voice box is ripped out by Blitzwing. However, the 2007 movie itself never establishes that it was Megatron who took Bumblebee's voice, only the tie-in comics which already have been contradicted by the other movies in many ways. As this is the only movie that in fact shows how Bumblebee lost his voice, it should be taken as canon over any previous statement, even if it is a retcon.
At the very end, as Bumblebee is seen from the front, driving across the Golden Gate Bridge, his headlights are on. When seen from the back, the taillights are not on - either they're on or off, but not both. However, the Transformer-copy of a vehicle, which is so different internally that it can morph quickly into a robot ; does not need to synchronize its front and rear lights.