In terms of engine work, the wide-body (A380, B747) fleet maintenance is done in Singapore and Malaysia, some of the domestic B737 work is done there. It's a bit more complex than that too because Lufthansa have a partnership with QANTAS but they do the work in Melbs so, technically, whilst it counts as 3rd party maintenance, it's locally done. Airframe work is almost all done locally at Avalon and Brisbane.
The issue is made even more complex because QANTAS get the manufacturer, Rolls Royce, to do maintenance too. That's considered by some to be iffy because if there's question marks over the engine itself, as with the Trent 900 (used on the Scarebus), aside from other vested interests, QANTAS are on the hook to keep using them as is and will be more so as B747's are phased out and the widebody fleet becomes exclusively A380 over the next 5 years.
Personally, the biggest issue facing airlines is crew training. The nature of experience is changing and whilst some would say automated cockpit systems make that a non-issue, I don't think anyone can be sure how it'll play out yet. Almost all airlines are going heavily towards the cadet route which results in pilots with a fairly homogenous flying experience (18 months full-time in lighties + sim time = ~300hrs ATPL). While the cadet programs result in new hires who fly with massively experienced captains with huge breadth of experience flying different types in different places and contexts, the centre will hold. Eventually, though, you'll have an airline full of pilots who've only known cadet-style training, captains with 2000hrs in command who've known no other flying than airline flying (fairly strictly controlled). They'll have never experienced getting themselves out of jail flying in a clapped-out C210 out of Kunnanurra trying to avoid inters which closed in on them or being forced to learn how to put down a dr killer in severe crosswind. They'll only know good flying conditions and sims. Again, with cockpit automation, some will say airline pilots won't need any of that but, of course, the one time in a million that you do (AF447?)......
Unrelated but the ATSB is releasing a final report into the Emirates scare in '09 this morning.
Emirates Melbourne 2009 near crash final report due today | Plane Talking