My son has the usual small, indoor caged pets: gerbils, birds. And I have a couple of tropical fish in a small aquarium. Now the birds, if they got loose, I'm not to worried if I have to clean up a deposit. But rodents chew. He's been warned. If they get loose and chew on an instrument, they're dispatched. Now, with that in mind, I keep my instruments in their cases, with the exception of my DB that currently resides on a stand until better arrangements can be made. And the house we live in has an extra bedroom converted to be a room just for my gear, with the door shut. So it should never be a problem.
Advice to all pet owners: keep your instruments in their cases and in a room that you can shut the door properly. This takes care of 99% + of all situations. If your pet can open the door, you've got bigger problems than this thread.
Also, I'm going to rant: I have no sympathy for owners who bitch about their cat spraying the cab or their dog gnawing the neck. You know better than to leave them out and around - either the instruments or the pets.