In the episode 'Carn-Evil' Cuphead says 'Aw heck' And also in the episode 'In Charms way' Somewhere in the episode Cuphead says 'What the heck was that?'
Some non-profane rudeness, and terms some would consider mildly rude, such as "ding-dongs."
None throughout all the show's first three seasons (actually produced as one, but released and marketed as three), with the closest being the use of the line "it really sucks an egg" in the episode, "A Very Devilish Christmas," and the direct use of "hell" in the description for a promotional clip for S3 on social media.
No profane but mild name calling/insults use mostly antiquated terms: "ding-dong," "oh, banana oil," "dumb yokels," "loser," "punk," etc. In season 1, "heck" is uses twice.
In "The I scream man", Mugman says "YOUR STUPID CROOKED HAT!" but crooked sounds like f---ing.