It's likely you have already figured this out: Landlords tend not to appreciate pets. If you have man's best friend, your hunt for a new apartment will be more challenging, but certainly not hopeless. Generally speaking, the more restrictive the apartment market gets, the more picky property owners can become, showing favoritism for tenants who don't have house animals, particularly agressive dogs.
Below are a few good tricks for renting with pets:
1) Most apartments accept pets to some degree. Restrictions generally center around the number of house animals you've got, aggressive breed dogs, or big pets.
2) Start by looking for pet-friendly rentals. Apartment complexes with dog-parks, or that offer various units with small yards, will probably accept more breeds or large pets.
3) Find some good references for your pet. Former landlords or building managers are best. Sometimes your vet might make a good reference. The point is for it to be crystal clear that you don't have a problem pet.
4) A lot of the opposition to dogs is a fear they'll bark often (particularly when the tenant is not home), disturbing other tenants. When your dog is not of the barking variety, make sure you point out that when applying for an apartment.
5) Some cat-friendly landlords will probably accept little dogs.
6) Pet cats have a much better possibility of being accepted if they are de-clawed, neutered and guaranteed to be housebroken.
7) Homes are more likely to take your pet, because they often have yards, so you may need to focus on that route.