I asked this before, but it was buried in a much longer post, and I got conflicting answers. So, just to pull out the essential bit:
If a character is mounted, and wants to be sneaky, who rolls the stealth check: just the rider, just the mount, or both of them?
And for purposes of getting the +20 vs the +40 bonus from invisibility: are both of them or only one of them considered to be "moving"?
(I would rule that some mounts automatically fail stealth, because horseshoes on hardpack is loud. But let's ignore that. In my scenario, the mount is a bonded animal companion monstrous spider in a cave.)
Sneak (DC 15): The animal can be ordered to make Stealth checks in order to stay hidden and to continue using Stealth even when circumstances or its natural instincts would normally cause it to abandon secrecy.
Your mount acts on your initiative count as you direct it. You move at its speed, but the mount uses its action to move.
I would say both rider and mount make stealth checks to remain hidden.
A creature larger or smaller than Medium takes a size bonus or penalty on Stealth checks depending on its size category:Fine +16, Diminutive +12, Tiny +8, Small +4, Large –4, Huge –8, Gargantuan –12, Colossal –16.
And the mount takes a -4 to the check.
(I would rule that some mounts automatically fail stealth, because horseshoes on hardpack is loud.
How about a hefty circumstance penalty instead?
If the companion knows the Hide trick (or something similar), then it's no sweat. Both mount and rider make checks.
If the mount doesn't have that trick, but could be stealthy if it had to, the PC needs to spend a little time "pushing" the companion to temporarily learn the trick. (see the Handle Animal skill)
If they aren't able to stealth at all, then that's it.