There are currently 5 wheels in the game as of the latest update (One was temporarily added in update 1.4.). Here are the six of them:
Old Giant Wheel (1.4 only)
This wheel is twice the size of the rest of the wheels. It is grey, it’s electric, and has an odd tendency to be bouncy. It can go through objects, ignore them (no effect) or ram them. It has since been removed, and thus never received any skins for it. See Smankusors' YouTube channel for more info.
A basic wheel that first appears in level 1-1. This wheel offers a method of moving heavy things more quickly, as it can roll. It can also improve the speed of non-wheel propelling items, such as the Fireworks and the Soda Bottle.
One-star variations include a blue wheel, a green wheel and a pink wheel.
Two-star variations include a wheel with spokes, a blue wheel with a picture of a duck, a red wheel with a picture of a duck, and a wheel made of gingerbread.
Three-star variations include a black wheel with a picture of a star, a red wheel with an unchanged mount, and a rainbow wheel. The red unchanged mount wheel looks light a doughnut
A 1-seater car powered by the Bellows, with a Umbrella (likely to stop the wheels breaking off, and halting the vehicle.)
Using a small wheel, behind a larger wheel, will incline the vehicle up.
A metallic counterpart of the Wooden wheel, the Metal Wheel is more durable, and it increases downhill speed. It debuts in level 1-8.
One-star variations include a bronze mount and a red mount.
Two-star variations include a black mount and a bone wheel.
Three-star variations include a golden wheel and a flower wheel.
A large car with different types of wheels.
Powered Wheels function similarly to a Metal Wheel. However, if it is part of a contraption with the pig or an Engine and is touching the ground, it can be activated to propel the vehicle forward. It can be made to drive the other way with the Lever. First appeared on level 1-29.
The Powered Wheel and the Metal Wheel share similar variations.
Sharing the power across multiple wheels can improve grip.
As its namesake suggests, powered wheels are engine-powered.
A smaller counterpart of the metal wheel, thus less likely to fall off, because of its size. This wheel is more often used behind the vehicle, mostly for planes, so that the plane is easily angled upwards, to lessen the time for it to take off. These wheels appear to be more springy than others, making it good for planes as it's springiness can enable easier landings with less breaks. First appeared on level 3-17.
One-star variations include a bronze mount and what is probably an eyeball.
Two-star variations include a pink wheel and a red mount.
Three-star variations include a golden wheel and a flower.
An alien variation of this part exists. This variation can be powered just like a Powered Wheel, though it's far faster. Interestingly, while the normal small wheel weighs less than the other wheels, it's alien counterpart weighs the same as the powered wheel.
The tiny wheel, behind the wooden one, will incline this vehicle up.
Tiny/Shopping Cart Wheel
The alien wheel is in the top right corner of this image.
Plunger/Suction Cup Wheels
A stickier counterpart of the Powwered Wheel, this wheel is able to move the vehicle onto terrain other vehicles could not without falling, such as a ceiling or vertical walls. In earlier updates, the wheel and the Plunger Gun's plungers would only stick to terrain, and would launch away player-placed objects (such as the vehicle itself.). This wheel is extremely important for the success of Rise and Swine and The Road to El Porkado levels. It first appeared on level 2-1.
The only one-star variation of this wheel has a bronze mount and blue plungers.
The only two-star variation of this wheel has a black mount and resembles an octopus tentacle.
The only three-star variation of this wheel is golden.
Plunger/Suction Cup Wheels
This vehicle is sticking to the rocks only via the plunger/suction cup wheels. Any other parts (with the exception of the Grappling Hook) will make it fall off.
King Pig, when placed on top of a Suction Cup Wheel, will fly into the air along with the wheel. (Patched)
An open umbrella placed next to a Powered Wheel will cause the vehicle to fly into the air.
This no longer works with normal umbrellas, and the powered variant must be used instead. It must also be closed and reopened manually, as the bug does not work with the umbrella in its default state.
Below are two videos explaining the glitch:
Bad Piggies Hoverbike by PIGineering 11 Nov 2012-1