Back to the Future

The audience learns the modern origin of the Tannen family's catchphrase insult, "Butt head." Biff Tannen's great grandfather, a Confederate officer in the Civil War, called his enemies "buttocks brains" but changed it to "butt head" after being corrected by one of Doc Brown's sons. That era's Tannen said, "Butt heads... I like the sound of it."

In the episode Marty wear the same cowboy suit from the Back to the Future 3 movie.

Verne starts to sing "53 bottles of pop on the wall" with is a spoof to the song "99 bottles of beer on the wall".

Marty makes a reference to Axl Rose, lead singer of Guns N' Roses

Jules makes a repair to the DeLorean - but the motor is in the front of the car. In real life (and in the three films) the motor is in the back of the DeLorean.

Where were Marty's family and Jennifer at the concert? You'd have thought they would have been there to support him and his band.

What happened to the rest of the Pinheads at the concert? Doc saves his family and Marty from the extreme weather, but the Pinheads are nowhere to be seen.

Doc apparently was born in 1922, making him 63 in 1985. But in the original trilogy he was 65 in 1985, making his birthyear 1920. Also, in the 2011 BTTF video game by Telltale Games, Doc was 17 in 1931, meaning that he was born in 1914, which would've made him 71 in 1985.

Comic books as we now know them didn't come along until the release of Funnies on Parade, the first modern comic book, in 1933. Before that time, newspapers were the main venue for comics in the United States.
There doesn't seem to be any obvious power source (like a Mr. Fusion) attached to Jules' spare flux capacitor for the barrel, leading to the question of how it generated enough power to activate time-travel. (Though I suppose Jules could've modified it somehow to be hydro-powered).

The Brown family visit 2091, in which they learn that the McFly family legacy has been immortalised in the McFly Space Cruises, in which Marty's great-granddaughter Marta McFly is a Captain.

When Jules and Verne fly the train through the big Courthouse billboard (the one that says "Historic Hill Valley"), the train flies directly through the clock. The hole is the same size as the clock. Yet in the next shot the hole has taken up half the Courthouse!

When Jules and Verne give the newspaper to the security guards, the front page with the story about the disaster is now in the middle of the paper.

Doc and Marty travel to South America in the 1500's to look for a toad that has since become extinct. They refer to the toad as Bufo marinus. In fact, this is the Latin name for the cane toad, which is far from extinct.

After they managed to get Franklin to do his kite expiriment and fix the past back up, wouldn't the chairs that Jules and Clara burned come back into existence?

When Franklin did his kite experiment, his son William was with him. But William Franklin is nowhere to be seen in this episode.

We see the Courthouse twice in this episode, and both times the clock is still working, not frozen at 10:04 PM like it should be!

Verne's name is spelt "Vern" on the scoreboard near the start of the episode.

Marty reacts to being called "chicken" by Bifficus and as such enters the race. Marty learnt his lesson at the end of Back to the Future Part III and should not react to being called "chicken" anymore.

Doc couldn't repair the train because he needed a flux capacitor - but he was somehow able to send the little truck forward to 1992 with the note for Marty.

Marty crashes into the barn door, creating a hole. But when he leaves for 1692 a few minutes later, the hole is gone.