1) If you Save and Quit your world while riding a horse it will vanish when you log in.

2) I was playing with a sibling of mine (WITH LAN) while riding my horse, I then decided to save the world. I expected to get back in the world with my horse.

3) However when I opened the world after saving it, while on my horse, I didn't spawn on it or anywhere else. I searched and searched but it was gone. i even replicated what i did with another world with no one on it, and the same result.

To replicate the bug
1. Turn on LAN
1. Spawn or find horses
2. Tame them and put a saddle on them as well
3. Ride around a bit
4. Save and Quit while on horse
5. Open World Again and Horse is gone