Half-horse and half-human, centaurs are a relatively common feature in classic fantasy fiction. They seem to be most suited as plains or steppes dwellers, but they are often placed in heavily wooded areas that would seem better suited to smaller varieties than are commonly described. There are some obvious issues with incorporating centaurs into urban fantasy, particularly in an human or city-based fantasy due to their size and shape. Some authors have played with this idea, though. Riordan developed a magic wheelchair for his key centaur while Eoin Colfer has a smaller breed of centaur that never leaves the faerie city, which has adapted vehicles and utilities for their shape. Rowling confined hers to a preserve in a relatively remote forest.
Centaurs are common in Greek mythology. They were noted for their violent tendencies and intoxication. However, the most famous centaurs include Chiron, the healer and trainer of heroes, as well as Nessus, who slew Heracles through deception. Greek myths also include the onocentaur, a goat variant that lives in the mountains, but which is less well known.
C.S. Lewis includes centaurs in Narnia, as allies of Aslan/good. His centaurs appear to be strong warriors, clad in barding and bearing large weapons to fight the forces that oppose Aslan and the Pevensies.
Rick Riordan brings in Chiron as a counselor at Camp Half-Blood as well as the paintball and root beer loving Party Ponies (virtually every centaur who isn’t Chiron).
J.K. Rowling’s centaurs are forest dwelling philosophers who have a talent for divination and astrology and a strong distaste for humans and wizards. They appear to be excellent archers, as displayed during the Battle of Hogwarts and their salute at Dumbledore’s funeral. They also hate giants.
Steven Brust, and others, have introduced variations on the “-taur” theme, such as his cat-centaurs in Dragaera (human top, feline below). Generally the variants serve roles as monsters, advisors, trainers, mentors, and warriors.
Often conflated, fauns and satyrs are two rather different entities, though they serve similar roles, particularly as male counterparts to nymphs. Both are goat-legged men and appear in rural environments. However that is where the similarities end. The Roman faun is a semi-civilized, perhaps domesticated, creature and generally has small horns. The Greek satyr is a wild, often intoxicated, creature generally devoted to Pan or Dionysus. The Greeks did introduce some female satyrs, but they were a late invention of classical poets and only had rare appearances. Neither faun nor satyr are especially common in fantasy or urban fantasy novels. They are, though, known to appear frequently in fantasy video games and in some comics and stories as lords of the woods or hills, often in quasi-druidic sex magic rituals.
Rick Riordan employs both in his Greco-Roman demigods series. The satyrs serve as guides and protectors for young demigods, bringing them safely (usually) to Camp Half-Blood and assisting in both guarding the camp and mentoring the demigods. They also act as male counterpart to the nymphs. The fauns come across more as meek servants and pitiful individuals in New Rome/Camp Jupiter, although they occasionally serve as nurses to aid injured demigods.
For being relatively minor figures in the fantasy and urban fantasy genres, gargoyles display a fair bit of variation. Usually they are grotesque or semi-grotesque humanoids. Some are living stone, others are not. They seem to be heavily associated with churches, which makes sense given the locations of real gargoyles, so there is a lot of potential as religious protectors, whether directly creatures of God or beings created by magically active priests. Regardless, gargoyles have a lot of potential as warriors, spies, and such in cities. In rural areas, they would be likely to favor mountains and cliff areas, as the closest natural equivalents to urban buildings.
The Gargoyles TV show presents them as “handsome grotesques” who turn to stone by day and become living, strong, flying beings by night.
Terry Pratchett introduces gargoyles to Discworld as a variety of troll adapted to urban environments. They are exceedingly patient and observant. They love cities and high places, where they are often paid in pigeons for information. Eventually a few gargoyles become official members of the Ankh-Morpork Watch, as ultimately happens with every species Pratchett introduced to the world.
In the Craft Sequence (a secondary world, post-industrial fantasy), Max Gladstone treats gargoyles as the creations of a, now believed, dead goddess. They were her elite servants, guardians, and warriors who protected her city (and eventually her remains). They appear human, but take a gargoyle form to fly, fight (stronger, claws), and meld with the city itself. They are fiercely devoted to the “deceased” goddess Seril and her surviving remnants. They are also feared by the city’s populace and hunted by its hive mind police force.