Took a Level in Badass: When he learns telekinesis. At first he fails utterly and only manages to hit himself in the head with a brick. When he gets angry though, he subconsciously masters the technique and effortlessly thrashes Cave Guy and Cobra Queen in battle. He doesn't use the ability in later episodes, though there were only two episodes left.
Too Dumb to Live: When Guitierrez wants to know his weakness, Freakazoid not only tells him what it is (a special cage) but helps him build it. In another episode he gets arrested on this charge by the Idiotic Police.
Younger than They Look: He's supposed to be the same age as Dexter (sixteen) but looks like he could be in his twenties.
Heart Is an Awesome Power: Cosgrove's ability to make ANYBODY immediately stop fighting and behave themselves, from common burglars to supervillain mooks. This might not sound too awesome since as a damn good police officer he can command authority...until you find out that he is the only person to have ever made the Warner siblingssit down and behave.
Parental Substitute: He seems to have a sort of fatherly role in Freakazoid's life; he takes him on day trips and is much more genuinely concerned about his well-being than Freakazoid/Dexter's biological parents.
Secret Keeper: In "Mission: Freakazoid" it's revealed that he knew all along that Dexter and Freakazoid were the same person.
Ultimate Job Security: Despite constantly using his police car for trivial reasons and wasting a lot of time going on odd trips with Freakazoid, he's still a well-respected police officer.
Bitch in Sheep's Clothing: Subtle example. Every once in awhile you can spot hints of a hidden mean streak. In "Dance of Doom" she lies to Dexter to get out of being his date to a school dance. She later agrees to a dinner date in "Dexter's Date" but makes him wait six months for it, all the while hoping he'd forget about it.
In "Mission: Freakazoid" she goes as far as calling Dexter creepy after finding out that he and Freakazoid are the same person. Freakazoid never calls her out on it.
Expy: The character of Professor Jones, voiced by Jonathan Harris, is essentially an animated version of Dr. Smith, Harris's character from Lost in Space. The show lampshaded this heavily in his debut episode by having at least 3 different characters asking him if he was on "that show with the robot". This was one of the reasons he was hired.
Fragile Flower: Will break down weeping or run away screaming in seconds.
Morton's Fork: He rather enjoys doing this; in "The Wrath of Guitierrez" either Freakazoid remains in the real world and has his powers drained, or he enters the Net where Guitierrez's Death Trap awaits. In "Hero Boy" he does something similar, except Guitierrez was threatening to vaporize Earth instead.