Flights to Sydney seem to be the most wide open with travel on Monday through Thursdays starting in mid-February through September with June, July, and August dates available. Most of these flights will be available on American Airlines with United matching on select departures.
Flights to Melbourne are just slightly cheaper, but the dates aren't as extensive. The travel days still consist of Mondays through Thursdays but the availability is only through early June, with far fewer available flight days in the spring, but if you are flexible with dates you should be able to make it work.
Why not see both cities on the same trip?! Multi-city options are open in the same range. Fly into Melbourne and out of Sydney or vice-versa for just about the same price!
There's a lot of fantastic fares flying out of O'Hare today so make sure to take a look at our Chicago (ORD) departure page for a full listing.
Don't live in Chicago? There still might be an awesome Australian adventure in your future too. Just have a look at our Sydney (SYD), Melbourne (MEL), and Brisbane (BNE) pages to see what's available from your local airport.