Mounts

Summary
Here are all the ones I can think of so far:

Wyvern riders: Melady's wyvern is called Trifinne, Zeiss's wyvern is Rubley, Vaida's wyvern is Umbriel, Heath's wyvern is Hyperion, Cormag's is Genarog. Percy's wyvern is named Michalis in the JP version and Ace everywhere else, Cherce's wyvern is named Minerva, and Camilla's wyvern does not have a canon name, but is commonly called Marcia or Marzia (though Marcia is a pegasus knight.)

Pegasus Knights: Sumia and Cynthia's pegasus is named Belfire, Tana's is Achaeus, Farina's is Murphy, Marcia's is Apples, Cordelia's pegasus is Aurora, Fee's pegasus is named Mahnya (the name of Fee's aunt), Vanessa's Pegasus is named Titania, and Florina's pegasus is Huey

Sophie's horse Avel is the only named horse in the series, based on the cavalier Abel.

Pegasi

 * Aurora (Rider: Cordelia)
 * Bellfire (Rider: Sumia, Cynthia)
 * Achaeus (Rider: Tana)
 * Apples (Rider: Marcia)
 * Mahnya (Rider: Fee)
 * Titania (Rider: Vanessa)
 * Huey (Rider: Florina)

Wyverns

 * Minerva (Rider: Cherche)
 * Marzia (Rider: Camilla)
 * Trifinne (Rider: Melady)
 * Rubley (Rider: Zeiss)
 * Umbriel (Rider: Vaida)
 * Hyperion (Rider: Heath)
 * Genarog (Rider: Cormag)
 * Michalis/Ace (Rider: Percy)

Horses

 * Abel (Rider: Sophie)