In addition to the continuity issue with the shotgun, the revolver carried by the constable (a Smith & Wesson .38 special) changes a couple of times also. Sometimes it is shown to be a standard tapered barrel revolver then at others, it's a "bull-barreled" or "heavy barrel" model.
Davey stepped in Mrs. Clarendon's blood when he first entered her house after her murder; however, when Mike Andersen returns to her house for the second time that night, you can see there is no footprint in the blood on the floor.
Even after it starts snowing heavily, not a single driver has their vehicle lights on until it's almost full dark.
Mike Andersen didn't photograph Mrs. Clarendon's walker when he returned to her house after she was murdered; he just brought it in the house.
Mike Andersen covered up Mrs. Clarendon after taking a single photo of her dead body. He should have known not do do this, as it would potentially disturb evidence of her murder.