It is different everywhere, but in Australia we seem to have:
Service - Eg: Guide dog, hearing assistance dog, physical assistance dog
Working - Eg: Police, search & rescue, airport sniffer
Support - Emotional support dogs
Alert - Eg: Diabetes, epilepsy & more recently I have read there is a move to allow proven migraine alert dogs.
Therapy - Those licensed & qualified to visit places such as hospitals, nursing homes, schools, rehab clinics & even prisons.
I don't know the specifics of your area but I believe the qualifications for emotional support dog can range from a simple note from your doctor to a full mental evaluation test as well as a display proving the dog does in fact aid you during times of need...and I dont just mean feeling a little stressed or down - I mean serious anxiety disorders, phobias, PTSD, chronic depression etc.
Unless you feel that you genuinely need a dog for my above mentioned reasons, please do not attempt to have him/her registered as a support animal. My best suggestion to you is to train your dog exceptionally well. No destructive behaviors, barking, anti social behaviour, aggression...& go for you canine Good Citizen Cert & prove that the dog is obedient.
Yes, it can be very difficult to find new accommodation but prior planning & good training can make the path a lot easier for you
Best of luck.