Finding a pet friendly rental has always been challenging, especially with large breed dogs, generally anything above 40 pounds. The best way is simply to read the entire ad and see if they are accepted and call for clarification. However in recent years more than half of the renting population are pet owners and the rental owners have recognized this is a significant part of their possible rental pool of prospective "customers". As a result of this demographic change most rental owners are accepting or at least considering pets. If in the advertisement for the unit you see "submit" when it comes to pets then at least the owner is somewhat open minded. I would suggest bringing your dog with you assuming the dog is well natured and has been taken to "relieve" themselves PRIOR to seeing the unit. I would also recommend bringing, just in case, copies of vaccination papers. Bringing a second person to "handle" your pet is a good idea while you take the time to talk or negotiate with the owner or agent away from the pet and can avoid unfavorable situations.
Deposits are another issue, some owners charge a separate deposit from $500 to $2,000 and some will simply include the pet deposit in the security deposit. There is no clear ruling mandating how to handle this specific event.
Hope that helps!