Some heroes, such as Batman, Spider-Man and Wolverine, name themselves after, or have powers based on, a member of the animal kingdom. Others actually are members of the animal kingdom, or at least the Funny Animal kingdom.
The result of bringing together the most popular action-cartoon genre with the most popular comedy-cartoon genre. Usually come in one of three flavors. One is a Funny Animal comic that is a direct parody of superheroes, and likely to consist of bad animal puns. Usually considered non-canon in most comic "universes." Another type often features Animal Sidekicks and even superhero's pets, and this often is canon, though unlikely to play a very important role.
And the last type, possibly the best known, are straight-up superheroes who just happen to be animals, often of the very anthropomorphic type. You can thank the Teenage Mutant Ninja Turtles for this type being so ubiquitous.
Will often have Alliterative Names. Not to be confused with an Animal-Themed Superbeing.
Comic book people sure love animal puns, don't they?
There's also the puntastic Marvel Apes series along with the Prime Eight spin-off. Just a small sampling: Spider-Monkey, Iron Mandrill, Nick Furry (same name as the funny animal in the Spider-Ham stories, oddly), Silverback Surfer, Doc Ook...
Does Black Panther turning into a literal anthropomorphic black panther in Earth X count?
Boris The Bear, an ultraviolent ursine Anti-Hero who's made it his life's mission to destroy every other anthropomorphic animal superhero in existence, starting with the TMNT, except for Droopy.
Not really. That was only in his first issue. Still, the fact that this exists shows you how popular, almost tediously so, this trope was in the wake of the Turtles.
Extinctioners - an indy comic featuring an entire planet of super-powered, human created "humanimals".
A few stories had Tom Strong meet his Funny Animal counterpart, a rabbit named Warren Strong along with his wife Patience, daughters Topsy, Turvy and Fluffytail (a Shout-Out to Peter Rabbit) and archenemy Basil Saveen (a fox version of Paul Saveen, possibly named after Basil Brush).
Rex, The Wonder Dog. He killed dinosaurs with atom bombs.
The DC version of this (in the Silver Age, at least) was the Legion of Super-Pets, set up in the 30th century (even though most of the members came from the 20th century!) as an adjunct to the Legion of Super-Heroes. The LSP consisted of Krypto the super-dog, Streaky the (sometimes) super-cat, Comet the super-horse, Beppo the super-monkey, and Proty, a shape-shifting blob of protoplasm that the similarly-powered Chameleon Boy had as a pet. Proty (and his successor Proty II) was the only Super-Pet native to the 30th century; just as well all those super-animals could travel through time, wasn't it?
And then there was the Space Canine Patrol Agency, an LSH-like team of super-powered dogs that Krypto encountered on one of his romps through space and joined, sharing a few adventures. They don't quite qualify for this trope, though, as all the members were depicted as life-like dogs rather than anthropomorphic animals; that was left for the villains! We will mercifully pass over the Space Cat Patrol Agency, who also appeared in one panel of the first SCPA story...
DC also has Hoppy the Marvel Bunny, a lapine version of the original Captain Marvel. Hoppy's "hero" name is Captain Marvel Bunny.
While Hoppy lived in an Alternate Universe full of talking animals, Captain Marvel's best friend in his own world was Mr. Tawky Tawny, an anthropomorphic tiger. Tawny was originally an aversion of this trope, since despite being a tiger, he was consistently treated as a more-or-less ordinary civilian. Modern stories have usually let him be more of an active hero, using his tiger abilities to help the Marvel Family fight evil.
At least in Italy (but probably, elsewhere), it's easier to list characters that have not a super identity. Donald Duck and Goofy are the most prominent, but even Goofy's nephew, Daisy, Josť Carioca (actually, a parody of superheroes), Gyro, and many others. It's good to know that at least once the Beagle Boys have been supervillains (with the X-Men powers).
These stories are now published for the English-language market by Boom! Kids as Disney Hero Squad: Ultraheroes.
DC One Million had Justice Legion Z (for Zoomorphs), which was descended from the Legion of Executive Familiars, which in turn was the even-futher-future counterpart to the Legion of Superpets. The JLZ membership included Proty One Million, a version of Comet the Super-Horse and Mastermind, presumably a heroic descendent of Mr Mind, the World's Wickedest Worm. The Executive Familiars were Krypto-9, Wormhole, the Sun Dragons, and Octo the Eight-Dimensional Cephalopod.
Howard the Duck, who is an accomplished martial artist and even knows limited sorcery; although Howard would probably take offense at being labeled a "super hero," since he considers super heroes to be just one more ridiculous element of the world he's stuck in.
Rocket Raccoon, and for that matter, a good number of his friends from his original mini-series.
Back in the Golden Age, Sheldon Mayer drew one Red Tornado strip with Ma Hunkell as a mother hen, Scribbly the Boy Cartoonist as a horse, and the Cyclone Kids (Scribbly's kid brother and Ma Hunkell's daughter) as a colt and a chick. According to a somewhat Self-Deprecating announcement at the start of the story, this was just to keep things interesting.
Independent and Underground comix have them too, naturally: QT Bunny and Wonder Warthog, to name just one for each. (Although calling Wonder Warthog a superhero is somewhat stretching the term...)
The Man-Eating Cow from The Tick comics. Who is actually a cow. Who eats people.
Cats & Dogs features superspy felines and canines, including Siamese "ninja", the Russian Blue (a cat assassin with a Russian accent), and Tab LazenbyVoiced by George Lazenby himself.
The 1979 movie C.H.O.M.P.S. is about a super-powered robot that looks like a fuzzy little dog.
In the world of Soon I Will Be Invincible, it's stated that the thousand or so superpowered beings include three dogs, four cats, and a bird, as well as more unspecified aquatic superbeings, which may or may not be human.
Live Action TV
Max the Bionic Dog from The Bionic Woman was a German shepherd who had served as the prototype for the bionic implants later used on Steve and Jaimie.
The Worlds of Freedom sourcebook for Mutants & Masterminds features a Funny Animal version of the Freedom City setting called Furrydom City. The Furrydom League includes Captain Thunderkat, Bunny Liberty, Miss Corbie (a raven version of the Raven) Duck Daedalus, Sea Otter, Johnny Rabbit, Bowmoose, and Dr Metropomouse.
The "Supertoon" setting for the Toon role-playing game.
Palladium Games' Teenaged Mutant Ninja Turtles and Other Strangeness, a system for creating mutated or otherwise genetically-altered animal superheroes.
The Super Sheep weapon from Worms. It's a sheep which flies around wearing a red cape. And then explodes.
The Adventures of Dr. McNinja brings us Sparklelord, a unicorn taking the form of A motorcycle the Doctor found, who is one of the few things the King Radical fears, because Sparklelord is an Omnicidal Maniac who nearly destroyed the Radical Land.
Bad Horse from Doctor Horrible's Sing-Along Blog is a rare example of an Animal Supervillain. He's also one of the few to be played by a real animal, albeit only in one short scene. Naturally this has turned him into something of an Ensemble Dark Horse, pun very much intended.
The Krypto series also featured examples of "Animal Supervillains", such as Ignatius, an iguana pet of Lex Luthor; Isis, one of many felines owned by Catwoman; Bud and Lou, a pair of hyenas belonging to The Joker; the Penguin's trained birds, and the Darth Vader-esque Mechanikat.
Plucky Duck of Tiny Toon Adventures took on a few superhero alter-egos, most notably as The Toxic Revenger and as Batman parody Batduck, with Hamton as sidekick Decoy. The latter set-up spun off a segment where all the major animal characters appear as Justice League members. Babs also parodied Supergirl once.
Plucky and Hampton also once tried to join an in-universe Animal Superhero team. The Radioactive Immature Samurai Slugs. They named themselves after painters as well—Earl Schieb and Sherwin Williams.
Although it's debatable whether Super Cow has actual super powers or is just Cow speaking Spanish and willing to use the proportionate strength of... a cow.
She's been shown flying on several occasions, so it's more than likely superpowers. In fact, it was made a plot point in one episode that her powers come from her "magic cape", which is actually Cow's blankie.
An obscure and very low-quality animated film called The Adventures of the American Rabbit stars a rabbit who fights forest crime by painting himself as the American flag and putting on roller skates to fly around and give speeches of Eagleland grade-A quality.
Super Chicken was a millionaire playboy/superhero chicken, living among humans in Pittsburgh with Fred, his dim lion sidekick.