This list is not exhaustive but does give you an idea of the areas laws pertain to so you can do more research. 

American-wide
The USA has the ADA (Americans with Disabilities Act) that has overriding laws about Service dogs in the US. 

Retailers are allowed to ask 2 questions:
1. Is that a service animal?
2. What tasks does the dog do to mitigate your disabilities?
 
In order to be considered a SD, a dog must be trained to do tasks that specifically mitigate your disabilities.
 
Here is the act info:
 
Here is Service dog related info:
 
 
 
 
 
There is no certification required federally so do not fall for fake certifications that charge money to “register” your service dog. 
If you do decide to certify your dog, then the only ones that are valid require you to do a live in-person test WITH your dog. You are tested as a team. Assistance Dogs International has schools that are accredited by them that do offer certification. They are recognized internationally. I have not found an organization will certify owner-trained dogs that they have not been in training with for a period of time.

Airline Carriers Act
 
In addition different laws govern Airlines regarding SD.
 
Housing

Fair Housing Act covers emotional support animals (which are different from service dogs) but your service dog may be covered under this law.
 
Internal Revenue Service (taxes)

Fees that can be deducted in the US relating to Service Dogs

State Specific Laws

And each state has different laws that cannot supercede those laws, they can only make them more accessible.
In California for example, Service dogs in training are allowed in public places. In some other states they are not.
 
This is the most current state by state comparison of service dogs laws that I have found. 


Please Note 

These are provided for general interest only. You will need to verify each is current and that the details are correct with a laywer if using them to fight a legal case. SDTI accepts no liability or responsibility for posting them here.