All of the links provided are to official government websites or embassies. The Turkish, Maltese, Cypriot and Chilean websites do not mention the working holiday visa specifically, so I have linked to the embassy contact details. A bit of information about the Cypriot visa can be found here and I also have a copy of the latest application requirements and form here .
Australians: Belgium, Canada**, Chile*, Cyprus, Denmark, Finland, France, Germany, Hong Kong, Ireland, Italy, Japan, Malaysia*, Malta, The Netherlands, Norway, South Korea, Sweden, Taiwan, Thailand*, Turkey*, United Kingdom, USA.
* = these countries have extra entry requirements, such as tertiary education, a letter of approval from the Australian government or language proficiency. For more details, see the Australian Immigration website.
** = Australians can apply for a second working holiday visa after the first one expires if they still meet all the criteria.
Trans-Tasman Travel Arrangement (wiki entry)
Since 1973, Australians and New Zealanders have been able to move freely between their two countries, living and working. There are a couple of industries that do not employ outside of their own citizens (e.g. national security), and there is a long waiting period to be eligible for welfare benefits.