Picking the right ESA for you:
Dogs
Both small and large breeds make good ESAs, however, it is important to do research beforehand. ESAs are meant to emotionally support someone whom usually is mentally disabled or struggles to do daily tasks because of a disorder. Because of this fact, not all breeds may be a good fit.
For example, extremely energetic breeds such as Border Collies, Jack Russells, and Boxers may be too much to handle for someone who struggles just to get out of bed in the morning. Another example are dogs with high maintenance standards, such as poodles, shih tzus, and sheepdogs. These types of dogs need regular professional grooming, and for someone who may struggle to keep up with their own hygiene, this may be more of a burden to deal with.
I wouldn't completely rule out any dog breed, however, these are just things to consider because we don't want dogs to be neglected no matter their owner's condition.
In the end personality is the most important thing. If you get an adult, pick one with a calm personality that already has some training experience. If you go with a puppy you need to be commited in training them. An ESA dog should be calm, well behaved, and quiet in public and at home. I highly recommend looking up Vitoria Stilwell for training tips and tricks.
Cats
Cats are fairly aloof creatures that may or may not suit the needs of someone looking for an ESA. Cats can be extremely affectionate and playful and very in tune with their owner's needs. However, some prefer their alone time and sleep much of the day. Sometimes simply the presence of a feline is enough for someone looking for emotional support. However, others need more of a active participant. In these cases it may be a good idea to adopt a cat that is already an adult, so you have an idea on whether they are more extroverted or introverted. If you are looking for a kitten, pick one based off what personality you most desire and do your best to mold them to your needs. Keep in mind, despite your best efforts, a cat will still be their own independent self as they get older.
In general, any cat breed can service for an ESA. Cats are not as tainted by humans meddling with their genes, so they are all still very similar in expectations. There are a few things to consider: a long haired cat needs daily brushing, a hairless (or nearly) cat needs regular baths, and flat-faced cats can suffer health issues. However, these are problems most commonly faced in expensive pure-breds, meaning you should be fine going to your local shelter and picking whatever cat clicks with you. When it comes to domestic cats, calicos have reputations of being sassy and orange cats are usually laid back; however, they are all individuals.
If you're interested in a cat being your ESA you most likely already love cats just the way they are. Either way, Jackson Galaxy has amazing educational videos on how best to work with your cat. Your ESA cat should enjoy being pet and held, doesn't destroy furniture, and uses the litter box consistently. Some people like their ESA cats to be harness trained to take them places, however, sometimes even the best cat doesn't like going to new places. So keep in mind that may not work out no matter how hard you try.
Other
Any animal can be an ESA because there is no species restrictions. The main thing to keep in mind is that the few laws that protect ESAs may not apply depending on the animal you choose. Be sure to look up the specific rules associated with the Air Carrier Access Act and Fair Housing Act before committing to a type of animal. As stated with the other selections: ESAs should be calm, quiet, and well behaved; but most importantly, provide emotional support. Always do research on an animal before adopting one.