Jack Griffin has been invisible through the course of the film. As such, it would have been impossible to see his reflection in a mirror and therefore he couldn't groom himself. Yet when he becomes visible in the final scene, he is perfectly clean-shaven and his hair combed.
One of the tricks used to suggest Griffin's invisibility was the simple use of black cloth to hide his exposed flesh. This is particularly evident when Mrs. Hall barges in on the Invisible Man while he's eating. The black cloths covering the lower part of his face and his wrists as he holds the serviette up are clearly visible.
The goof items below may give away important plot points.
A vengeful Griffin vows to kill Kemp the next night. This is followed by a daylight shot of Griffin pushing two searchers off a cliff, then a night shot of him derailing a train, followed by a daylight shot of him looting a bank and distributing the money outside to people passing by, indicating at least two days must have passed before keeping his "promise" to Kemp.
The hospital doctor says the effect of the invisibility drug will die along with Dr. Griffin (the invisible man). Since the invisibility drug is unknown to anyone, there's no way for him to make such a "knowledgeable" statement.
In the last scene in the Lion's Head bar, the Invisible Man lifts a bench to kill Inspector Bird. The bench was actually lifted by wires, and the technician(s) who operated them apparently misjudged the distance when they brought it down on the prostrate man. It appears to hit him on the nose - hard enough for him to reach for the injured proboscis before dropping his hand as he "dies".