Ann brings home a new battery-powered radio, turns it on, and music immediately begins to play. Radios still used vacuum tubes in 1948 and those would need a few seconds to warm up before the unit could produce sound.
The calendar seen in Tom's apartment when the murder occurs in July begins on a Monday, which necessitates that Tom's execution date (first Tuesday after Christmas) was December 31st aka New Year's Eve. Doubtful.
The flashback sequences that make up much of the film are set up as Tom remembering how he ended up hours from electrocution on Death Row, but many of the scenes in the flashbacks show events Tom was not present at and would have no likely way of knowing about.
The whole investigation revolves around finding a shoe impression in the mud outside. The police conclude that the murderer left it, so finding the owner of the shoe means that one finds the murderer. But the impression was outside where ANYBODY could have made that impression, including any number of innocent passers-by who moved on elsewhere, as well as the murderer. Such was what should have been called a RED HERRING in murder mystery parlance.
When the detectives are looking at the dripping faucet and the shoe print in the dirt, the detective says, "By the time they get through, we'll know what he looks like and everything about him. It's better than a photograph." While he's obviously exaggerating, this is a ridiculous statement. A lot more effort is needed to get all that information from a shoe print (e.g., determining each store where the shoe is sold and then going there, questioning the sales people), whereas a picture would pretty much nail down the suspect just by knocking on a few doors.
Police detective Judd detains a murder suspect, Kosloff. During this detainment Judd has Kosloff follow him a couple of times. But NO police officer would have a murder suspect follow him. This would give the suspect the opportunity to overpower his captor. If not handcuffed, the police officer would have at the least had his suspect go first where the police officer could keep an eye on his suspect.