After Alice fills out the form to see the Inspector, she lays down the pen on the table. In the next shot she still has it in her hand.
The signature on the painting Alice blots out is not the same one she put there originally. It is much smaller and the underline is missing.
Alice grasps the knife in a normal cutting position, with her thumb at the front of the handle. But her grip is different as she takes it behind the curtain: the knife is now held below her fist, with her thumb at the back of the handle, in a stabbing position.
In the sound version of this film, the drawing of the female figure that The Artist drew (under Alice's cartoon drawing of a head) in his studio differs from what was displayed at Scotland Yard, due to stock footage from the silent version (which started production earlier) being used.
Alice's hair is different on multiple occasions on switching camera angle close-ups. Most noticeable when Crewe is attempting to assist Alice button the back of her dress...and in Scotland Yard right before Alice and Frank stroll down a long hallway.
When Alice "unlocks" the door to the building where she lives, it starts to open as soon as the key reaches the door. It was clearly not only not locked, but not even latched. However, she goes through with the motion of unlocking it.
At about 0:24:30 when Crewe (Cyril Ritchard) is talking to Alice (Anny Ondra), he calls her "Anny" before correcting himself.
Mr. Crewe never lights the fire after he asks Alice if she feels cold.