SQ is not a ZED member and you'll need to get ID tickets for their flight (provided your airline has agreement). I'm not sure how ZED works in US (I've heard people say its a blank ticket that lets you fly within the tickets range) but in Asia, you pretty much need your ZED tickets specifying the exact port.

If you are coming from SFO, suggest you do CX as HKG-SIN is quite frequent everyday so that'll be SFO-HKG-SIN. I don't like CX as I find their seats a bit cramp (not good for tall people) but its just a few hours that you have to struggle with. I find myself flying them more than I want to mainly due to timing.

Date wise avoid Oct 1 to Oct 12 week as its a holiday in China/HK/SE Asia ( a bunch of various public holiday falling during these period) also thanksgiving week. Otherwise there's not much holiday that will fill up the seats