I've flown via a number of different routes and now know what my favoured options are.
The quickest way I have been was via Japan on Japan Airlines. I cannot remember the flight time - it wasn't short, but considering you fly straight over the north pole it is shorter than say Singapore. On the way out I stopped at Osaka for a few (not very restful) hours. On the way back I stopped at Tokyo overnight and flew the next day - much better.
The worst way was to fly via Greece with Olympic Airways. It was a truly awful journey and the way back almost put me off ever flying again. Part of that was down to the awful quality of the airline but the sheer amount of travelling and tedious stoppages was exhausting. And it still didn't eliminate the long haul flight aspect as the flight from Athens to Bangkok was longer (IIRC) than the flight from London to Osaka / Tokyo. It also included a 7hr stop in Athens Airport, without doubt the worst major international airport I have been to and overall the travelling lasted for 30 hrs which nearly finished us off. The return journey was even worse as we had three stops instead of two and the airline was awful. Just don't go there!
The best way, without doubt, is to break your journey. The last couple of times I've opted to go with Qantas via Singapore, with a scheduled 48hr break in Singapore. This gives you time not only to recuperate but you get to have a mini holiday as well. On one visit to Singapore I even managed to fit in a day trip to the Jahor Baru area of Malaysia which was interesting and acquired another stamp for Mr Passport - even if my multiple entry and exits of Malaysia / Singapore did cause a furrowed brow at Australian Immigration. I swear Australia's immigration officials are a special breed. Their countrymen are so friendly and laid back but immigration officers are a scowling, growling affair - I swear they must breed them on special ranches where they get poked a lot to make them angry the whole time.
I keep meaning to write a site about my stopover trips in Singapore but I haven't yet - so you'll have to take my word for it!
Hope this helps!