When Barry leans forward to fall into the pool from the diving board (ala Cameron in the movie), the chair he's sitting on falls with him into the water. A few seconds later, when Beverly comes running to the pool to save him, the chair is back on the diving board.
According to the narrator's voice-over at the beginning of the show, this takes place in the spring, yet there's a homecoming football game. High-school football is only played in the United States from August through December, and cold-weather locations often end their seasons in November.
The Scene where Barry in Bed faking illness, below the "RUN DMC" poster changes from the "Ferris Bueller" poster to a dark colored poster when Erica storms off and finally back to the "Ferris Bueller" poster.
While The Goldbergs takes place in Philadelphia, after the Fiero is stolen (mimicking the scene from Ferris Bueller), when the Fiero is depicted flying over a crest in the road, palm trees and other Southern California vegetation are clearly visible.
In the narration, adult Adam says that Pops' car, the Firebird, was never seen again. The car was a Trans Am, not a Firebird.