Corrected entry: When Billy is talking to the Sheriff and his Deputy, he explains what the creatures are, and shows them Gizmo. The cops don't take Billy's story seriously and even mock him, even after the phone call about the Futterman's Snowplough "accident." Why didn't Billy simply tell the Sheriff about the dead Teacher or get them to call the Dr where Billy left his injured mother? The mere mention of a dead body should have to be investigated by the police, and the word of a local respected Dr should also be enough to back up Billy's story and give him the help he's need to stop the Gremlins.
The Rodent
2nd Nov 2015
Correction: The sheriff and deputy were seen after talking to Billy simply driving away from a man dressed as Santa being attacked by gremlins; right before they got into an accident (and presumably why they were nowhere to be seen the rest of the movie). They are drunk and incompetent. Billy not doing a good job of convincing them of Gremlins is not a plot hole, but a plot contrivance.