The meeting to nominate the delegates. Especially "The Bar's Closed" Running Gag.
Peabody ranting to himself drunkenly in his office, even down to talking to the jug.
For Black Comedy: When he returns to the office and continues rambling...and lights the lamp to reveal Valance in crew.
"Liberty Valance taking liberties...with the liberty of the press?!"
"I'll get it, Liberty!" Cue Tom kicking the Mook away.
When the Doc is shown Valance on the street lifeless, he calls for "Whiskey!" like it's a medical order. And then he drinks it.
During the statehood convention, the ranchers have a staged rally to humble the homesteaders, complete with a horse-riding sign waver storming onto the stage. While the rider puts on a rodeo show much to the judge's anger, the horse quickly takes a drink from the judge's water pitcher.
As the rancher's political stooge stands there smiling during the lasso stunt, one of the homesteaders shouts out "Now that you got a rope around him, hang the horse thief!"