Revealing mistake: When the camera is tracking in to Peter Lorre whistling in the cafe, part of the screen gets noticeably darker because the camera is blocking one of the lights. This happens in other scenes, noticeably, when the camera makes shadows.
Continuity mistake: The second-story room of the prison warden's daughter is on fire and she is screaming for help. Stan has trouble connecting a fire hose to a water valve and Ollie winds up completely drenched, but in all the following scenes where Ollie is using a ladder to try to rescue the woman, his clothes are dry.
Continuity mistake: Near the end, when Tom Powers is waiting in the rain to murder Schemer Burns, we see Tom wearing a tall, straight-crowned fedora that is only slightly tilted to his right. Suddenly, and for no apparent reason, we see a close-up of Tom wearing a completely different, taper-crowned fedora that is tilted deeply to his left. This close-up lasts for 7 seconds, then the shot returns to Tom wearing his original tall, straight fedora.