Verses in the Book of Mormon that talk about "horses"

Horses associated with travel and chariots

Alma 18꞉9-10

And they said unto him: Behold, he is feeding thy horses. Now the king had commanded his servants, previous to the time of the watering of their flocks, that they should prepare his horses and chariots, and conduct him forth to the land of Nephi...Now when king Lamoni heard that Ammon was preparing his horses and his chariots he was more astonished...

Alma 18꞉12

And it came to pass that when Ammon had made ready the horses and the chariots for the king and his servants...

Alma 20꞉6

Now when Lamoni had heard this he caused that his servants should make ready his horses and his chariots.

3 Nephi 3꞉22

And it came to pass in the *seventeenth year, in the latter end of the year, the proclamation of Lachoneus had gone forth throughout all the face of the land, and they had taken their horses, and their chariots, and their cattle, and all their flocks, and their herds, and their grain, and all their substance, and did march forth by thousands and by tens of thousands, until they had all gone forth to the place which had been appointed that they should gather themselves together, to defend themselves against their enemies.

(It should be noted that we are not told if these chariots served a purpose in riding, or if they were for transport of goods, or if they had a ceremonial function. One assumes some sort of practicality or ritual use in war, since they brought chariots to the siege in 3 Nephi.)

Horse mentioned in quotes of Old World scripture

2 Nephi 12꞉7

Their land also is full of silver and gold, neither is there any end of their treasures; their land is also full of horses, neither is there any end of their chariots.

2 Nephi 15꞉28

Whose arrows shall be sharp, and all their bows bent, and their horses’ hoofs shall be counted like flint, and their wheels like a whirlwind, their roaring like a lion.

3 Nephi 21꞉14

Yea, wo be unto the Gentiles except they repent; for it shall come to pass in that day, saith the Father, that I will cut off thy horses out of the midst of thee, and I will destroy thy chariots;

Wild horses

1 Nephi 18꞉25

And it came to pass that we did find upon the land of promise, as we journeyed in the wilderness, that there were beasts in the forests of every kind, both the cow and the ox, and the ass and the horse, and the goat and the wild goat, and all manner of wild animals, which were for the use of men.

Domesticated horses

Enos 1꞉21

And it came to pass that the people of Nephi did till the land, and raise all manner of grain, and of fruit, and flocks of herds, and flocks of all manner of cattle of every kind, and goats, and wild goats, and also many horses.

Horses as a potential source of food

3 Nephi 4꞉4

Therefore, there was no chance for the robbers to plunder and to obtain food, save it were to come up in open battle against the Nephites; and the Nephites being in one body, and having so great a number, and having reserved for themselves provisions, and horses and cattle, and flocks of every kind, that they might subsist for the space of seven years...

3 Nephi 6꞉1

And now it came to pass that the people of the Nephites did all return to their own lands in the *twenty and sixth year, every man, with his family, his flocks and his herds, his horses and his cattle...

Ether 9꞉19

And they also had horses, and asses, and there were elephants and cureloms and cumoms; all of which were useful unto man, and more especially the elephants and cureloms and cumoms.

It is interesting that the horses are often grouped with cattle, and seem to have played a role in the diet (though this may have been under the exigencies of the siege of 3 Nephi.)

The Book of Mormon never mentions horses pulling chariots or being ridden

A few things to keep in mind:

The Book of Mormon does not mention horses pulling chariots. The BOM does not mention horses being ridden. Horses are mentioned with chariots several times. Assuming that they were present in order to pull the chariots must be extrapolated.
The Old Testament and New Testament do mention horses being ridden. The D&C mentions that horses can be ridden.
Joseph knew much about horses yet in the Book of Mormon, they are not used in any way he was familiar with. They are not mentioned as being used for work, transportation or battle.
Joseph likely knew, as everyone did, that the European horse was introduced by the Spanish. Why, then, did he make such a clumsy error in his forgery?
Critics of the Church falsely attributes the possibility of the word "horse" as a description of a similar animal to Joseph mistranslating the text. No one claimed that Joseph "mistranslated" the term deer for horse. The accurate position is that early Nephites may have labeled deer "horses." This conjecture is based on the fact that The Amerindians called horses "deer" when they first saw them.

Notes

Verses in the Book of Mormon that talk about "horses"