My husband and I have rental properties and advertise usually as either NO PETS or SMALL DOGS ONLY.
We absolutely positively will not rent to anyone with cats. Don't even try talking us into a cat!!! I love them tremendously, but know first hand the destructive nature of cat urine!!
Dogs however are a different story. We will bend on the dog rules if the prospective tenant is upfront and persuasive in advocating for their pet. Puppy? NO WAY (again, first hand eperience with the chewing and Potty accidents).
Big dog who is older, house trained, crated while mom/dad is away - we could be persuaded. Key is how you sell yourself and your pet.
Offer to pay a pet deposit and $20-25 extra rent per month. Offer to let the landlord meet your pet and make periodic inspections after you move in. Let the landlord know that your pet is treated on schedule to keep fleas away, and that you pick up after your dog.
Small landlords, like my husband and me, fear having expensive damages that we can ill afford. Someone who is apparently sensitive to our worries and willing to go to extra steps and expense upfront to diffuse the fear of possible damages, scores major points with us.
Apartment complexes usually won't be too flexible about pet size, but my son rented an apartment off of New Centre Dr. and I know some tenants had huge dogs. Can't remember the name of the complex though.
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Be a passionate advocate for your pet and your sincerity just might do the trick.