When MacPherson (Dana Andrews) goes to Lydecker's (Clifton Webb), he is nosing around with some of his art stuff and picks something up. Lydecker tells him to leave it alone or some such, as it is priceless and not to break it. However, we will see that Lydecker is in the bathtub, and the door is nearly completely shut, he could not have seen Dana Andrews doing anything from that vantage.
Just before Laura returns to her apartment, McPherson is holding a drink and falls asleep in an armchair. The camera pans down and the glass is visible in his hand resting on the arm of the chair. As the camera pans up his empty hand is seen on the armrest.
MacPherson tells Laura at one point that the shotgun retrieved from her country house has been ruled out as the murder weapon (and thus potentially clearing Shelby). Shotguns fire pellets and cannot be tested ballistically like bullets, thus it would be impossible to rule it out as the murder weapon.
Lydecker recognizes McPherson's name from "the siege of Babylon, Long Island", where McPherson had gone in and captured a criminal in spite of being shot. However, McPherson is a New York City detective. Babylon is in Suffolk County, Long Island, some forty miles outside the city, and well out of McPherson's jurisdiction, so it makes no sense that he would have been in a gun battle that far away from his area of authority.
The clock, which is heard to strike the time, only has one keyhole on its face. However, because the striking mechanism is separate from the timekeeping mechanism, a chiming clock must have at least two separate keyholes - one for the clock itself, and one for the strike. Presumably the prop was made only to keep the time since the production staff knew they would be adding the sound of the chime in post-production.
Lydecker tells Det. McPherson that Laura called him the night of her murder to say she was leaving to spend a few days at her cottage. Yet that same night Lydecker goes looking for her at her apartment. However, as Waldo was lying about the night of the murder, it is possible that he simply forgot what he had said before.
When McPherson questions Laura after her return to the city, she tells him that no one had seen her when she'd gone away for the weekend, that she'd been out of all contact with people and news the entire time, and so had never learned of the murder. But she later tells him she had asked her handyman to repair her radio as she was leaving town. Therefore she'd seen at least one person since the murder had become news, and in her small town, the handyman would almost certainly have heard of the murder and been able to tell her about it.