When Lucy and Ethel are on the roof and are handcuffed by the policeman, the handcuffs are on Ethel's left arm and Lucy's right arm. In the next scene, Lucy opens the apartment door and yells in to Ricky and Fred for help, that they are being arrested. She opens the door with her right arm, which is not handcuffed.
When Lucy and Ethel try to get the boys' attention by going to the corner store and telephoning them, Lucy gets her coat and Ethel's coat from the closet. Ethel just lives across the hall - she wouldn't have put her coat on to go to Lucy's apartment.
When Lucy cuts the electrical wire on the rooftop with a pair of scissors she would have been electrocuted.
When Lucy gives her phone number to the sergeant at the police station, she gives a Murray Hill number. However, the Murray Hill neighborhood stretches from 34th Street north to 42nd Street, while the Ricardos are previously stated to live in the east 60s.
When Lucy and Ethel run out of the drugstore, with the policeman in pursuit, their shadows are visible on what is clearly a backdrop outside of the door.