Riddles -- common
Short-form puns -- commonly attempted
Gracie Allen joke generation -- interpret a text said in one context in a different one, e.g., "I need to get to work. Have you seen my car keys?" "No, but since you're in a hurry, ou can show them to me when you get home."
Long-form puns ("shaggy dog stories") -- the puns are not hard to generate but stories are; for people, it's the opposite