I just booked a SQ F award through LH, NONE of the one-stop F flights to the U.S. had seats available for about 30 different dates I asked. SQ 25/26 FRA-SIN-FRA is usually never available either through LH. I got SQ 325/326 FRA-SIN-FRA, these always had 1 O-seat available but never two. Had SQ 325 last time as well, so that shouldn't be too difficult.

You may want to look into booking LAX-NRT and NRT-SIN or SFO-HKG and HKG-SIN separately (the same flight as 2 sectors), that way there should be some availability. SIN-BKK-SIN will not be a problem. Or indeed IAD-FRA-SIN - SQ 325 and LH 776 leave at about the same time from FRA, so maybe you could secure the 1 O seat and get the other ticket first confirmed on LH776 while waiting for SQ to open up another O seat.

That was my experience with LH, UA might of course have different availability. My dates were 3-8 months out.