When Igor, Inga, and Dr. Frankenstein go to catch the monster for the first time, they try to inject him with a sedative. Before she injects him, she squirts a little out the top to make sure no air is in the needle, but when she sticks him with the needle, she does not inject anything. In fact, when she pulls the needle back out, she squirts more sedative onto the ground.
When the policeman confronts the chained-up Creature in the jail, he pulls a cigarette from behind his ear, grabs a match a lights it. He notices the Creature doesn't like fire. When he grabs a second match to scare the Creature, his cigarette is back behind his ear again.
Throughout the film, Frederick is referred to as the grandson or great grandson of Victor Frankenstein. However, Victor was a character from 1818 when the novel was first published, which would make Frederick in the 1930's Victor's great-great-great-great grandson, minimum.
The candle that operates the bookcase door appears to go out momentarily, and then comes back on, before Frankenstein and the assistant go down the passageway, but it simply got caught in a draft and dimmed completely down before coming up again - normal behavior of candles.
When Gerhard Falkstein meets Dr. Frankenstein in his class, he refers to Baron von Frankenstein as his great-grandfather, whilst in the rest of the film he refers to him as the grandfather. However, it is stated later in the film (most notably in the deleted sequence "The Reading Of The Will"), that the Frankenstein who's will is being read is the Great-Grandfather, and the infamous Victor Frankenstein is his son, or Frederick's Grandfather.
When the criminal is being hanged, it's raining heavily. When he's buried, the gravediggers are shoveling dry, almost dusty soil onto the grave. When Frederick and Igor dig him up, the mud on their clothes and the coffin is soaking wet and water can be seen dripping down near where Frederick had been lifting. This may have been intentional for Igor's "could be raining" line.
Throughout the movie, Kemp uses his left hand to move his prosthetic right, but as the villagers are looking for the Monster through foggy forest, Kemp uses his right hand to adjust his left, complete with sound effects. Probably deliberate.
When Frederick and Inga are caught after having sex, the shot of the platform coming down from the roof is reused from when the Monster is reanimated. You can see Dr Frankenstein's legs and lab coat standing on the side of the platform as it lowers.
In the 'Puttin' on the Ritz' number, one of the electric footlights explodes, alarming the Monster. The light however, stays entire and is still on, and it isn't clear what it is that Frederick is stamping out.
In the scene where they are removing the casket from the ground, they appear under the casket to lift it out. That would be impossible to do unless they dug a huge hole with a way to get under the casket.