Am I allowed to take my existing pets to the shelter to meet my potential new pet to check they get on before adopting?
Should I ask the shelter first, and if so what do I say?
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You should definitely ask before just showing up with a pet in tow. The answer will probably depend on the type of pet.
My local shelter arranges "meet and greet"s for dogs, so it's probably possible for your dog to meet a potential adoptee dog. Cats really don't do well out of their homes (and I notice that they don't have feline "meet and greet"s). For bunnies, I'm pretty sure they actually require a supervised introduction if you already have a bunny. This is, of course, just anecdotal evidence from one (US) shelter, but it's meant to illustrate the range of possibilities.
As another answer notes, reputable shelters will allow you to return a pet who doesn't integrate well into the household. But that's a sad and frustrating situation for all involved, so if they can allow you to find out earlier about inter-pet issues, within the limits of health and their facilities, it's in their interest to do so.
If you have concerns about introducing your existing pet(s) to the potential pet(s), you should definitely talk to your rescue group/shelter. For some pet species, an introduction on neutral territory is helpful and can be a good indicator of how well they will get along. For other species, removal from the home environment can be so stressful that you will get no useful information (and may harm the beginning relationship).
Additionally, if you have an existing pet who is difficult to get along with, I recommend looking for a rescue organization that does extensive home fostering and personality matching.
Many pets don't behave like themselves in a shelter environment with lots of other animals, noises, strangers, and other general stressors. An animal like that does best in a home (foster) environment where it can relax and show its personality. Then, when you contact the rescue and describe your existing pet, they can look through notes kept by the current fosters and find someone who is a likely personality fit. It's a longer process (especially if you want a specific color/sex/other attribute), but it tends to work out better in the long run.
The rescue group I've worked with will occasionally do "test runs" where they will let you take the cat home for a period of time since they aren't sure how it will react under certain circumstances.
Finally, the rescue should instruct you on how to integrate your new pet and be available for advice during the process. The rescue group I've worked with suggests allowing at least 6 months for integration of cats. Sometimes it takes longer (we kept Ginger locked alone in the bedroom when we weren't home for a few years before she learned to get along with everyone).
You can ask the shelter, but the answer may be "no". They may require proof of vaccinations, lest you expose other animals to illness.
However, reputable shelters will let you return a pet if things don't work out. And there's often an adjustment period, so just because two animals don't get along that well when first introduced doesn't mean they won't be great friends in a week. The shelter will advise you about the best way to handle the introduction.