SQ has a direct flight between SIN and SFO, in fact a couple (one via HKG and one via ICN). If you can get into business class the flight via ICN is better - since it has the newest business class seat.

I second the suggestion to check out the alliance forums for an around the world (RTW) ticket. There is plenty of info on pricing, routing options, tips for best places to start to save some money, etc.

If you do need to fly in economy consider something like LHR-BKK-SIN (on Thai) or LHR-HKG-SIN (on Air NZ and SQ) due to Thai and Air NZ's extra seat pitch, then SIN-LAX-SFO (on SQ LAX nonstop flight which is premium economy then UA), then SFO-LAX-AKL (on United and Air NZ) due to NZ's extra seat pitch again.