I entered the USA on a three month visa waiver program from Australia. I am about to return to Australia and would like to come back again to the USA. How long do I have to wait until I can re-enter the USA for another 3 months?
There is no official rule on this. It is up to the discretion of the customs office dealing with you at the border to decide if you can or can't enter. This applies whether you have a visa or wish to use the VWP.
The prime directive of the officer is to assume you wish to live in the USA. It is up to you to prove otherwise. The amount of "proof" required may be as simple as your verbal statement that you intend to do XYZ whilst there, and intend to leave on XYZ date. But if the records show you have been spending more time in the USA than you have outside the USA, the officer may decide you are misusing the VWP system to "live" in the USA.
My personal experience is that some ports of entry have officers with significantly stricter attitudes. Others are more lax. For instance, in my experience, the difference between entering the USA via Florida and New York is like night and day. I've entered through both on many occasions. Coming from Australia/New Zealand, I've also noticed a different between LAX and SFO. I always come and go through SFO if possible. It's not as much of a difference as MIA and FLL (both in Florida), but enough to warrant avoiding LAX. Also the airport is less busy in SFO so that's a plus.
It is my understanding that if you are denied entry you will likely require a visa to return to the USA (from memory, this will hold true for a period of 10 years, but it may be indefinite, I am not 100% sure). Receiving a visa may also be more difficult if you were refused entry at the border. Caution is advised.
I think you can enter again the day after you left. I remember entering again about 100 days after I entered the first time (so not long after the end of the 90 days, but a while after I left the country) and I have friends who entered a lot of times and on a regular basis so they probably had maybe 2 weeks between the time they left US (and gave their I-94 back) and they entered again (filling a new I-94). We were all European and part of the Visa waiver program.
On an official website no other condition than having a valid passport (and the ESTA procedure if you arrive by plane) is required.
Be careful, though, to do the ESTA procedure if you arrive by plane and to give I94 back when you leave the US (if you got it when entering - it is the green paper with these weird questions)