You are able to live and work in Dubai as long as you have a valid UAE residency stamp in your passport. In order to get this you need to either be sponsored by a company (for which you work), or by your husband (if you have one). The residency stamp is valid for three years after which it is renewed depending on the type of employment contract you have.
As a non-UAE national you will never be allowed permanent resident status. There is a mandetory retirement age of 60 (or 65, sorry can't quiet remember), after which your Residency is revoked have to leave the country.
Work may or may not be easy to get, depends on the type of work you do. As an Australian you need to have a terssiary education and be fully qualified in some facet. Your papers need to be attested by the UAE consulate in Australia (proberbly in Sidney??)
There are plenty of Australian living and working in Dubai, like fleas on a kangaroo! Austrade has office here as well as some of the local Australian authorities.
I think you need to have worked here for 3 years before you get a Tax Exemption Certificate - until then you still continue paying tax in Australia.
Dubai is quiet different ... I would suggest you speak to as many people who have actualy been and stayed here - not tourists. After the first 2 weeks the shine wears off!
If you have any specific questions - drop me a mail. I cannot find work for people though!!!!! That is something you need to do yourself!