Emotional support animals are different from service animals, which are often trained to help blind people, etc. In most cases, people with severe anxiety disorders and other mental health disabilities require emotional support animals. In general, landlords that otherwise do not allow pets will be required to allow the emotional support animal as a "reasonable accommodation" under the Fair Housing Act.
Both the landlord and the tenant can lose a lot of time in a dispute. If a tenant keeps pets when the lease specifically forbids this, the landlord can initiate eviction proceedings. Or if the lease is silent on the topic altogether, conflicts may arise between different tenants due to noise or damage caused by the pet, or their may be no recourse for damage caused by the pet other than use of the security deposit.
It's no secret that tenants with pets tend to stay longer. As mentioned, you may have the upper hand compared to landlords and places that don’t accept pets and tenants are usually willing to pay more if you allow them to keep pets. Even though this is not always the case, most tenants with pets are highly responsible, or else it could be hard to convince future landlords to allow them to keep pets.
Start with not leaving your pet home alone for too long. Cleaning up after your pet is showing appreciation to your neighbors and landlord. Also, it is a good practice to introduce yourself to new neighbors when moving in and explain that you have a pet and that they should just let you know if a problem comes up.
Some landlords choose to charge a pet fee. States like California only allow a fixed amount of security deposits and pet fees are forbidden. In the less-regulatory states, you can ask for a pet fee to account for the additional wear and tear that some pets are certain to inflict on your property.