Equidae

Horses and donkeys have never been shown in a Fallout game, but have been mentioned on occasion.

Background
Some equidae survived the Great War though none are directly encountered.

According to the Fallout Bible, Pugsley was an example of a mule (offspring of a male donkey and a female horse) that survived, though he is referred to as a donkey in-game.

The dialogue with Ethyl Wright mentions the widespread availability of "jugs of fermented mare's milk" (from a female horse) for any tribesman or tribeswoman. In order for a mare to produce milk, it generally must be four years old to reach sexual maturity and complete the gestation period of about 11 months, after which it will produce milk for its foal for four to six months.

Horses also appear in the graphic novel All Roads. An alleged email with developer Chris Avellone surfaced in 2013 regarding the appearance of horses in the comic, but has not been independently verified.

Post-War

 * The aforementioned Pugsley was a donkey owned by a man named Irwin who was killed when his ranch was attacked by raiders around 2161. As this was 84 years after the Great War and donkeys do not live longer than 50 years under even the best circumstances, it is probable that small populations survived on a domesticated basis.
 * Dane, one of the Children of the Cathedral, will mention horses in his insane ramblings.
 * A "Horse Marker" can be found in the G.E.C.K. This is a leftover from The Elder Scrolls IV: Oblivion, similar to the unused "Rumors" function in quests.
 * The Chosen One, when talking to Ethyl Wright, can mention the abundance of mare's milk being around among tribesmen.
 * There is a book entitled Pretty Pretty Horsies: A History of the Mongol Empire available in Fallout: New Vegas. It is a quest item for convincing the Great Khans to break their alliance with Caesar's Legion, by revealing the history of their namesake ancestors.
 * In the graphic novel All Roads, an NCR soldier seen in Chance's flashback of the Bitter Springs Massacre is riding a horse.
 * In the Fallout: New Vegas add-on Honest Hearts, the war clubs used by the Dead Horses have been carved into the shape of a horse head.
 * There is a drawing of a horse, chicken, and cow used in a framed picture near Bunker Hill in Fallout 4. The same drawing appears at an unmarked trapper outpost northwest of the MS Azalea, in the add-on Far Harbor.
 * Scout Leader Jaggy laments about missing the sight of horses in Appalachia.
 * Horses are referenced in the description of the Ol' Painless weapon in Fallout Shelter.