I like the coffee shop scene. In an interview with James Whitmore (on TCM), he complained that John Huston told him to "turn up the radio just before the cops enter."
Whitmore, who plays Gus, the owner, was afraid his lines would be drowned out by the music from the radio. I think it was a great touch. Sterling Hayden slips onto a counter stool and silently hands over his gun. Gus slips it into the cash register smoothly, silently, no questions asked.
Obviously, he's done this before.
Then, when the two big cops storm in, Gus stands up to them, insisting they "Get a warrant."