Ironic that you should ask this question when the National Dog Bite Preventions Week 5/19 -5/25 starts on Sunday. Every year, dogs bite more than 4.7 million people, according to the Centers for Disease Control and Prevention (CDC). Here is a link to the Humane Societies with information about preventing dog bites: http://www.americanhumane.org/animals/stop-animal-abuse/fact
You are not alone in having a difficult time finding a rental with a dog. Landlords have a right to descriminate against pets, smoking and poor credit history. We have seen the need for rentals increase since the housing crisis has caused so many homeowners to lose their homes. So there is a lot of competition for a descent rental. That being said, if you were a landlord and you have more than one application, everything on the applications is similar, number of occupants, credit history, etc, BUT one applicant has a dog and the other one doesn't, who would you choose?
Landlords do not want to take the chance of being sued. Many have had bad experiences with pets that have damaged their properties inside and outside. Most landlords will say right up front whether or not they allow dogs, cats, etc. That greatly limits your choices. So if you are working with an agent, they will not show those listings to you.
One thing you may want to consider is purchasing pet insurance. When submitting you application show proof that the dog has been spayed or neutered and that you have insurance. Also, get a reference from your current landlord that recommends you and your dog. This addresses the issue right up front.