Continuity mistake: As the crazy Mogwi changes from a ball to an actual creature, a can suddenly changes position on the shelf above, though nothing touches it.
Continuity mistake: When Billy and Kate return to the Clamp building as the Gremlins hatch, look closely - Kate is wearing beige/tan colored pantyhose/tights in the first scene of them running to the building. ("How long do they take to hatch?", "Not long enough, come on!") Almost immediately after arriving, Kate is told to begin her duties as a tour-guide, and thus doesn't have time to change her outfit. However, in one of her next scenes, she is suddenly wearing brighter cream-colored pantyhose/tights. (And given her fairly consistent screen time as a tour-guide, it's clear she didn't have time to change this one part of her outfit).