I would have voted for two options if I could have: yes, and it depends.
I'd have to meet the owner and the dog together and see the dog in various situations. (Hanging out in front of the property, etc.) I wouldn't base a yes or no solely on how it responded to me. I would want to see the dog as children played, bicycles went by, etc.... I would want to see how the owner handled the dog and the control they had. Let's face it, a ten pound dog can't pull an owner over -- a poorly behaved large dog most certainly can. I'm not a small dog fan, but it is just true. If a potential renter came to me with a large dog, it had BETTER be very well behaved to be considered. Yes, I'm giving small dogs more license there due to the above. Now, if the small dog was viscious and presented any threat of biting, that dog would get the ix-nay also.
Pretty obviously, not too many landlords want to go to that extent to do all those checks.
I'll go against the grain and say yes, if it is my property, I would likely automatically ban certain breeds, but I *might* be willing to give them a shot if they could pass various tests as stated above. As a property owner, I must limit my exposure to lawsuits, etc. If certain breeds would force my insurance rates higher, I wouldn't hesitate to ban them. Simple business.