I had been playing 1.8 for several hours before this started happening. Now it doesn't seem to go away. (see attached screenshots)
It started happening after launching the game and loading my hardcore SSP world.
The horses sometimes render normally, but when I look in certain directions or stand in certain positions they appear glitchy, until I look somewhere else or they turn/move.
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MC-67608 looks very different from what I'm seeing. It looks like the geometry is heavily distorted, whereas in my case the geometry is normal but sometimes renders behind more distant entities.

Please reopen.

Is there an invisible spider nearby? (see MC-79697)
If this is not the case, can you please attach the world as a zip folder?

Please remove the incorrect duplicate link. This is very likely caused by a bug which disables depth but does not re-enable it again like MC-79697
However this report is incomplete, see also this reddit comment of the reporter

@Marcono1234 Yes, there is in fact an invisible spider nearby. It's in that cobblestone enclosure in which I had trapped it.

alright, resolving as duplicate then

... well it even says "Invisible Spider Bro" on one of the signs. I should probably have had a closer look on it