It depends where they want to have a stopover as to which airline they can fly with.
I flew to Sydney with Singapore Airlines and had a two night stopover in Singapore. Another time I flew with Malaysia Airlines to Adelaide and had a stopover in Penang on the way there and on the flight back from Brisbane a one night stopover in Kuala Lumpur, If you fly with Thai Airlines you can have a stopover in Bangkok, Emirates in Dubai, Korean Air in Seoul, Japan Airlines in Tokyo, Cathay in Hong Kong, I'm not sure about BA and Qantas, maybe Hong Kong or Singapore. I have probably missed one.
Basically the airlines allow you to stay for one or more nights in the City where you would have to change planes (there are no non-stop flights all the way to Australia) without any extra charge. The airline can often book hotels at discount prices and some offer free tours and transfers to the city (i.e. Singapore Airlines). Malaysia airlines offer a free flight anywhere within Malaysia plus transfers to the hotel if you are flying in and out of Malaysia with their airline.
I haven't booked a stopover up to now on the internet, the last twice I booked by ringing the airline but I think that you can probably do it by using a search engine like Expedia.

Thanks so much for the quick reply Judith. I think i'll take your advice and phone the airline company.

I think they want to go with Singapore even tho' korean airline has a better price

thanks again Judith

karen

Karen
They would change planes at Singapore if flying with Qantas.

I have recently found flights on the Emirates websiteto go to New Zealand, and the FLight Centre have matched them. THey organised a three night stopover at Dubai on the way and one night in Sydney on the way back. It was slightly more espensive having stopovers as you get charged extra tax if you leave the airport apparently.

Has anyone changed their itenery once leaving the UK? My small print on e-ticket says that changes can be made after departure for free - does this mean that en-route I can change my flight back from Auckland to Sydney by one day to give us a couple of days there and arrange this when in Dubai when we stopoever on the way to Auckland?? Hope this makes sense. We are flying Emirates.

You may get a better response if you post in our Flights forum.

luci

Thanks Luci, have tried again on the flights forum.