Whether or not a greyhound is required to wear a muzzle in public is different depending what state of Australia you live in. Here is the round up.
It is ONLY those greyhounds adopted as pets from GAP, and wearing the green GAP identification collar, that are exempt from being muzzled in public. ALL other greyhounds, whether they are current or retired racing greyhounds or greyhounds adopted as pets privately or from a shelter or rescue group, MUST by law be muzzled in public.
Reference GAP Victoria
Despite the fact that GAP greyhounds are not racing anymore, it is currently still law that they wear a muzzle in a public place unless they are wearing their GAP approved green collar. The person responsible for the dog can be fined if they are unmuzzled and not wearing the green collar that clearly identifies the dog as having been through the GAP program.
Reference GAP Queensland
New South Wales
One of the main stumbling blocks that exists for the Greyhound Adoption Program in NSW is the requirement that our beautiful dogs be muzzled in public places. The negative effect of this stipulation of the Companion Animals Act is twofold:
It has caused the completely erroneous perception to become entrenched, in the minds of the general public, that Greyhounds are dangerous and aggressive.
In itself, the legal requirement that they be muzzled in public, discourages a lot of people who would otherwise be keen to adopt one of these gentle and aristocratic dogs.
We strongly believe that this legislation is misguided - not to say draconian - in that it, quite unjustifiably, discriminates against Greyhounds. (The only other breed which is specified in the CAA is the Pit Bull Terrier). It should be noteworthy that the Greyhound does not rate a mention in any published statistics of dog bites, despite the fact that far more Greyhounds are whelped each year than any other breed.
Furthermore Australia is the only country, with the sole exception of Northern Ireland, which requires Greyhounds to be muzzled when they are off the track.
Happily, Victoria amended its legislation in 1999 to allow GAP Greyhounds to be exempt from wearing muzzles. Since that time, the number of retired Greyhounds adopted in Victoria has increased more than threefold.
GAP NSW has been actively campaigning to get the current legislation amended to allow Greyhounds, who have been through the GAP program, to be exempt from wearing muzzles.
Reference GAP NSW
Please also be aware that although we are trying to change the situation, the current dog act also requires greyhounds to be muzzled and on a lead when in public. We have been strongly lobbying for the Greyhounds who go through the training in the Greyhound Adoption Program Tasmania Inc. to be allowed to go un-muzzled in public as we believe the dogs have been temperament tested and are well socialised.
Reference GAP Tasmania
I couldn't find any definitive references except for a discussion on Dogs Online - I believe the NT laws are the same as WA & NSW.
Australian Capital Territory
The ACT fixed their muzzle laws at the same time as Queensland did. Same deal with a green collar program.
Reference GAP NSW
Under new law, GAP dogs in South Australia no longer need to wear a muzzle in public after they have passed their appropriate assessment by the Cat and Management Board. Once they have passed they will be issued with an identification number and a Green GAP collar. Fosterers are to ensure that all dogs currently in foster need to wear their muzzle until they have passed their assessment and have the appropriate authority to be unmuzzled.
Reference - GAPSA
According to a law introduced in Australia in 1927 Greyhounds must always wear a muzzle when in a public place. Initially the muzzle was introduced on the race track to prevent these valuable animals from injuring themselves during the general excitement at the track and their high spirited racing. We have been lobbying to have companion greyhounds made exempt from this law, as is the case in Victoria, because they are such a friendly, affectionate and loving breed and when wearing a muzzle people often incorrectly assume the opposite.
Reference GAP WA