Plot hole: Given it's a government building there's no way the ghosts could drill into the U. S. Mint and not draw security's attention.
Factual error: There's an exhibit in the Smithsonian Institute that shows Benedict Arnold, William Demont, and Major John Andre labeled as Famous American Traitors. One problem, Andre wasn't a traitor to America as he was a British officer to whom Benedict Arnold arranged to surrender West Point to during the war.
The Ghost of The Bad Humor Man - S1-E4
Continuity mistake: The back doors of the armored truck that the phantoms are trapped in go from being open to closed to open again.
There's A Demon Shark In The Froggy Dark - S1-E9
Deliberate mistake: Unless they're super strong, none of Mystery Inc could've thrown an anchor to the top of a cliff.
Scared a Lot in Camelot - S1-E10
Other mistake: Fred calls out to the Black Knight ghost who sees and hears him. Yet it's Shaggy and Scooby in different armor who enter the room.
The Harum-Scarum Sanitarium - S1-E11
Continuity mistake: Shaggy goes back and forth from being in the front seat to the back seat of the Mystery Machine as the episode starts.
The Harum-Scarum Sanitarium - S1-E11
Continuity mistake: Fred and Velma get chased by one of the dogs into the room where Shaggy and Scooby are. Daphne wasn't with either pair, yet she's suddenly there once Fred closes the door.
A Frightened Hound Meets Demons Underground - S1-E14
Character mistake: Velma mentions Seattle burned down in 1890. The fire went from June 6-June 7 1889.