Renting an apartment that will allow your dog can require a little more work, but it's definitely doable. Even if a landlord has had bad experiences in the past with a tenant's dog, you have a chance to show them what a responsible pet owner looks like. About.com gives you the lowdown on renting with a dog.
If you already own a dog and need to rent, be sure to mention your pet to potential landlords right off the bat. There is no use wasting your time with buildings that don't allow pets if you already have one. Once you think you've found the perfect place for you AND your dog, go over the pet policy portion of your lease with a fine toothed comb. Make sure you know all about pet deposits, cleaning fees and any restrictions that exist on the size of your pet.
If you are already renting and decide you want to get a dog, check your lease to be sure pets are permitted. Even if the lease doesn't allow pets, you might approach your landlord and ask, you never know. If your landlord gives the go ahead, make sure to get more details. What is the weight or size limit? Are puppies allowed? How much is the deposit and any additional monthly fees? What are the rules about where a dog can be walked on the grounds of the apartments, leash-free areas, etc? If your landlord says no, either don't get a dog or move to a rental that allows them. Never try to hide your dog, both you and your dog are likely to wind up homeless!
Finally, if you do get a dog, make sure to be a responsible pet owner. Don't leave your dog home alone for long periods of time so they start barking and destroying your house. Clean up after your dog when they go outside and only let them off-leash in designated areas. If your dog does tend to ruin your furniture and home when you leave them alone, crate training can save your belongings and your deposit.