You will fly out of Toronto, Canada (YYZ) via Newark, Chicago and Hong Kong to Singapore in United Economy and back. You WILL also need a strong stomach and no fear of seat mates for such an endurance test. But at 2.83 cents per mile, this is as cheap as it gets.
You will earn full miles with United Mileage Plus when traveling before March 1st 2015. That should be ~20,000 miles. I'd rather sign up for a credit card and earn 20,000 miles that way (I do not really care about airline status anymore) but if you are new to the game and want to see Singapore, then why not!