Are there different rules for particular types of dog?

Yes.

Prescribed Breeds

The Dog and Cat Management Act 1995 (SA) contains additional restrictions on prescribed breeds of dogs. Section 4 currently prescribes the following breeds:

Prescribed breeds must be muzzled and under the effective control of a person by means of physical restraint when not confined to the premises of the owner or person responsible for the control of the dog [s 45B(1)]. It is an offence to breach this requirement. Higher penalties also apply to offences under the Dog and Cat Management Act 1995 (SA) that involve a prescribed breed of dog.

Greyhounds

Greyhounds must be under the effective control of a person by means of physical restraint and muzzled (unless exempt from muzzling) when not confined to the premises of the owner or person responsible for the control of the greyhound [s 45C]. It is an offence to breach this requirement.

Attack trained dogs, guard dogs and patrol dogs

The owner or person responsible for an attack trained dog, guard dog or patrol dog must ensure that [s 45D]:

It is an offence to breach these requirements, with a maximum penalty of $5,000 (expiation fee: $750).