Stumpy and Mr. Cat usually get along pretty well, and work together in several instances.
Stumpy admires Mr. Cat and frequently acts as his sidekick. In the episode "Let's Play Dubbing", Stumpy uses animation software to give himself Mr. Cat's personality and admits that he has always dreamed of being just like him. In "Let's Play Shaolin Fitness", Stumpy takes martial arts lessons from Kaeloo to prevent Mr. Cat from taking him stuff, only to take over the cat's role as the villain after knocking him out. In "Let's Play the Quest for the Wholly Gruel", Stumpy mentions that he thinks Mr. Cat always has cool ideas. Stumpy seemed to be the most affected by Mr. Cat's disappearance in "Let's Play Goodbye, Mr. Cat!", as he was crying the hardest of all and screaming "Mr. Cat! Don't leave me alone!".
Mr. Cat is usually not very nice to Stumpy, and he makes fun of him a lot along with the others. When he gets annoyed enough at Stumpy, he beats him up or even threatens to attack him with dangerous weapons. Nevertheless, he does actually care about Stumpy. In "Let's Play Frosted Christmas - Part 2", Mr. Cat shows that he does trust Stumpy in times of danger. In the episode "Let's Play Stumpy's I.Q.", when Stumpy dies (but is later brought back to life by a spirit), Mr. Cat digs a grave for him and actually attends his funeral (something he normally wouldn't do if someone died). In "Let's Play the Very Special Episode", Mr. Cat shows genuine remorse for all the bad things he did to Stumpy.