The Cockroaches love stirring up trouble everywhere they go. They have even taken vehicles for joyrides, like a dump truck, train, firetruck, and planes ("Rock 'n Roll Altitude", "Panic in the Air", and "First Flight"). They additionally commit crimes willfully on a near-daily basis with no remorse whatsoever. Like numerous villains, the Cockroaches derive pleasure from seeing people suffer, whether it be physically or emotionally. It should be safe to say their favorite target is Oggy (who they also have a peculiar relationship with). The trio expresses no sympathy for the feline and become convulsed with laughter upon witnessing him being put through any degree of pain. On a lesser note, they are shown to be incredibly slovenly as they are prone to making messes. The Cockroaches are even stated to find Oggy's house "much too clean for their tastes." They also seem cognizant of the fact that they are in a cartoon; they sometimes look directly at the viewer(s).
The Cockroaches' goals vary from episode to episode. Sometimes, they seek to steal comestibles (particularly from a certain blue cat), but this is not out of malice; in these cases, they are just hungry and need to eat. In fact, they were so desperate for something to eat in "Oggy's Invention" that they were seen eating part of a wall and a wooden shelf. Then again, more often than not, the trio aims to ruin Oggy's life in some new cruel way; they practically live to make him suffer. In addition, they can be quite vindictive. There are even instances in which the trio has sought vengeance on Oggy simply for irritating them, rather it was intentional or not. (Though, to be fair, there are some instances in which Oggy has wronged the Cockroaches without motive; e.g., "Don't Barge In!".)
Despite their almost entirely negative history with Oggy, the Cockroaches secretly need him in their lives. In "Copy Cat" and "Oggy Robot," Oggy became so depressed that he did nothing but sit on the couch as if he was rooted to the spot. As a result, the trio grew so bored they decided to build a robotic clone of the feline which would act exactly like him. In the episode "Moving Out", the Cockroaches were initially ecstatic upon finding out that Oggy had moved out, passing it off as a joke. However, when they find out that he was truly gone, Joey and Marky burst into tears. In addition, Dee Dee and Marky insanely crushed themselves repeatedly with Oggy's fly swatter just to refresh their memories of the blue feline swatting them. When Joey discovered the location of Oggy's new home, the trio did everything in their power to reunite with him (which horribly failed).
As abrasive as the Cockroaches are, they truly care about each other ("Caviar on the House!" being the most notable example). They were also extremely happy to see their cousin visiting them in "Roachy Redneck" and "A Roachneck Cousin".