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The best bet is to consult a Vet before giving your animals spent grain, yeast, hops or other brewery trub.
For dogs, the issue is that in their brewhouse form, hops inhibit their ability to shed heat by panting. Since dogs can't sweat, they can't regulate their body temperature. Although you should never give an animal finished beer, the hops left in it aren't in a form that does the same thing.
That said; for dogs, you can make great treats out of spent grain and even the big chunks of malt proteins that come up near the beginning of the boil. Just skim this, blend with a little peanut butter and egg yolk, then dry it in the oven.
Horses are susceptible to colic from spent brewhouse grain. Not all breeds have this issue, but many do. That said, horses can digest fresh, dry malt and they love it. Living brewhouse yeast is a terrible problem for all breeds of horses.
Pigs have issues with brewers yeast, but can eat both spent grain and brewhouse hops.
Cattle seem to be able to digest all forms of brewhouse goods. They'll fight over spent grain, though I've seen some get angry over a lot of roasted malts. They don't like hops, but when mixed with enough pale grain, they eat it with no obvious ill effects.
Chickens and ducks can eat spent grain, but they really love when insect larvae are in it. Their yolks seem to be richer with a barley diet. Hops are not good for chickens, for the same reasons as dogs. Chickens don't seem interested in fresh yeast. Ducks can develop natural foie gras on an all barley diet, but more slowly than traditional methods and not as rich.
Rabbits love spent grain. Meat rabbits gain a lot of meat weight and fat on spent barley.
Goats: Oginme raises goats, so he is more expert than I.