The non-classical animals are more concerned with particular strategies and techniques, and not as much with an all-encompassing worldview of combat. Nonetheless, they include some very fine fighting systems.
Praying Mantis:
Praying mantis style is a very famous style, developed in the 1700s by a fighter
named Wang Lang. He supposedly developed it specifically to defeat the monks of
the Shaolin Temple. The story is that he had been a very successful fighter who
decided to test himself against the monks and failed miserably in his first
fight. He then supposedly devoted years of his life to developing a fighting
system with which he could defeat them. The result, we are told, is Praying
Mantis style (named, it is said, the praying mantis whose defeat of a much
larger cicada inspired Wang Lang to study its movements) The monks, in a pattern
that was repeated many times in history, adopted the resulting style into the
curriculum of the Temple.
Praying Mantis is a combination of a set of sophisticated deflections, counters, and grappling movements with Monkey style footwork (see Monkey style, below). The fundamental strategy of Praying Mantis is to wait patiently for an opening (often in the form of an attack), then tie the opponent's arms with a grappling technique and strike into soft areas and nerve centers.
Shaolin Bird:
Shaolin Bird style is one of the older fighting styles, being derived from the
very old Lo Han style by way of the later China Hand style that seems to form
the basis of much of the familiar Korean and Okinawan styles. (Many of the
movements in Okinawan karate and such styles as Tang Soo Do closely resemble
movements in China Hand and Shaolin Bird styles).
In Shaolin Bird style the hard, linear strikes and kicks of Lo-Han and China Hand first begin to acquire some of the circularity and fluidity that is characteristic of many later Chinese styles.
The strategic assumption is Shaolin Bird style is that the opponent is larger and stronger. The Bird stylist compensates by leaping in to deliver a flurry of strikes, and then leaping back out of range; or, again, by goading the opponent into a charge and sidestepping while striking. Bird style relies on quick transitions between low and high attacks and stances, sudden reversals of direction, long-range jumps to cover ground quickly, and well-developed stamina. Bird forms emphasize elbows and finger thrusts to soft targets.
Monkey:
Monkey style is an advanced style that demands much of its practitioners. Like
Shaolin Bird style, it assumes that the opponent is larger and stronger, and
compensates by making it hard to reach or hold onto its practitioner. The Monkey
stylist jumps, flips, rolls, and climbs to avoid his attacker. He attacks from
peculiar angles, and contorts his body to strike when the opponent believes
himself safe.
A monkey stylist, if faced with an opponent who likes the lunging attacks and strong stances of, let us say, a Shotokan stylist, might sidestep the lunge, climb onto the opponent's knee to elbow into the head, and then dive into a roll to escape retaliation. Monkey stylists strike with the backs of the forearms, with the elbows, and with hook kicks and ape kicks (like a front snap, but twisted inward to strike like a roundhouse, but with the leg turned the opposite direction). Monkey stylists like to tease their opponents into rash action and take advantage of their rashness.
Eagle claw: (yingjowpai)
Eagle claw style is an animal style derived from the grappling art of Shaolin
Chin Na. It relies on very powerful seizing, pinching, twisting, and locking
techniques to immobilize or punish an attacker. Eagle claw stylists work hard on
developing their grips to facilitate application of painful locks and nerve
pinches. Like jujutsu, Eagle claw employs leverage and joint manipulation to
defeat an opponent.
Other styles:
There are many more Shaolin animal styles. A suggestive list might include White
Ape, Wild Horse, 10,000 Bees, and Golden Centipede.
And I'm sure there are others...