Robyn - I'm in singapore all the time and there are only two ways on UA - thorugh NRT or HKG - both connections and at least three to four flights for you -- but would STRONGLY suggest you consider the parnter airlines as you might save a connection and quite a bit of time...

To be honest - If I were you I would reconsider Singapore -- there is much more fun in Tokyo or HKG than in Singapore and so much more history and culture - and the food is amazing. Singapore is a nice place but it is a bit sterile and to be honest not so much to do unless you like to shop... Enjoy your trip!