Continuity mistake: Before Elmer's two dogs shake hands on agreeing to catch Daffy, one of them draws a diagram which is a dotted line and a circle with a cross in it. After the two dogs run into a tree, they go back to their meeting point and this time the diagram is completely different to the one drawn before.
Continuity mistake: Before Daffy puts the hot towel on Elmer, you can see that Elmer has two hairs on his head. When Daffy puts the towel on Elmer and pulls off his head, the hairs have disappeared, and when Daffy puts Elmer's face back on his head, the hairs have reappeared.
Audio problem: When Elmer grabs his gun and says "Ooh, I should have known better!" his mouth doesn't move.