Magical traits

It seems that ponies and other creatures alike have some magical power, mostly unicorns due to their ability to wield actual magic.

Winged unicorns

 * Princess Celestia and Princess Luna possess magic at much higher levels than usual ponies. They can control the sun and the moon respectively

Earth ponies

 * Earth ponies might be able to hold something without touching them as seen in The Ticket Master as when Twilight was far from Applejack she was still holding the ticket without her hooves or mouth, and in The Cutie Mark Chronicles when Applejack is able to carry her things on a pole.
 * Earth ponies, since presumably more connected to the earth, might sense whats going to happen or what is there but this might be only to Pinkie Pie.

Pegasus ponies

 * Rainbow Dash in The Ticket Master did a move in her daydream called 'the buckeneer blaze' offscreen, seemingly to create lighting.
 * Rainbow Dash also created a Sonic Rainboom, a rainbow out of thin air from breaking the sound barrier.
 * When a pegasus flies fast or maybe even a normal pace a trail of what color their mane is or something else like The Wonderbolts have thunderclouds following them.
 * The ability to walk on clouds without magic seems to be inherent to pegasus ponies (barring exceptions like Lyra in Sonic Rainboom and one earth filly in The Cutie Mark Chronicles.
 * All pegasus ponies seem to have the ability to levitate things as long as they are in contact with it, such as the movers wagon in Feeling Pinkie Keen.
 * Pegasus ponies Control weather patterns or stray clouds.

Unicorn ponies
Unicorns have many abilities mostly some of them having to do with their talent
 * Unicorns can hold objects by picking them up, a simple thing
 * All unicorns may be able to teleport, under stress or having a lot of concentration.
 * Unicorns can preform a spell to make themselves and other non-pegasus ponies able to walk on clouds.
 * Unicorns can make wings appear on non-pegasus ponies but takes a lot of power to do.
 * Unicorns can make light shows and illusions, like Trixe and Rarity.
 * Unicorns can turn one object into another but it will turn back without enough concentration.
 * Unicorns may be able to magicly connect to animals as in winter wrap up most ponies waking up the animals are unicorns and the ones walking with bunnies are unicorns

Other Creatures

 * A cockatrice can turn living creatures into stone by staring at it.

Spike is able to teleport certain things with his fire to certain areas, such as with teleporting scrolls to Princess Celestia.
 * Celestia's pet Philomena can rise from its dead ashes.
 * Griffions, like Pegasus ponies, can also stand on clouds.
 * The Poison Joke plant can play little 'pranks' on whoever moves around in it.