Exterior shots of the entrance side of Lona McLane's apartment building show it to be angled, but the map of the building in the stakeout room has all straight sides.
When Paul calls Lona in her apartment during the stakeout, it is raining. When he meets her on the roof a few minutes later, it is not raining, and there are stars in the night sky.
The nurse went to Apartment 423, immediately next door, to ask for ice cubes, Around 1:13, when McAllister is dictating a phone message for Eckstrom, he says the woman who lives in 426 saw a man in 423 when she asked for the ice cubes. It isn't likely that she lives in 426, since she lives on the same side of the hallway, her apartment would have to have an odd number, i.e. 421 or 425. And, sure enough, at around 1:22 the nurse is seen entering her apartment door clearly labeled 421.
When Paul and Paddy are in the lookout apartment, Paul has his left arm leaning against the window and then takes it off as he turns to look at Paddy. However, in the next cut which is a two-shot, he repeats the same action of taking his arm off the window but this time he doesn't turn but looks over his left shoulder.
When Lt. Eckstrom explains the layout of the apartment building to the detectives, he points to the apartment where the stakeout will take place, then points to a room just below it and tells the detectives to be sure to avoid contact with the other tenants. The room he points to is a bedroom in the stakeout apartment, so no tenants would be in that room.
It's blatantly obvious that the women being watched by the police from the apartment across the way could easily see the cops if they just looked out their window.
When Sheridan looks down from the roof, a line of cars parked along a curb is seen. But the cars are parked bumper to bumper, revealing that none of them were parallel parked (there is no space between them to allow a parallel parking maneuver or to pull out). Instead, it is obvious that each one was driven straight in, one by one, into position and parked behind the preceding one.
Police (or anyone else) surveilling someone would never do so standing right behind a window holding binoculars so as to be observed by anyone across the way or outside the building.
As in Double Indemnity (1944), although Fred MacMurray's character is not married, he wears a wedding ring throughout the film.