Are you flying F, C/J or Y?
UA doesn't have code-share agreement with SQ, so you would need to book SQ coded flight if you want to fly SQ.
And, honestly, there is absolutely no advantage of purchasing codeshare flights.

I normally fly UA if purchasing Y and upgrade to C using SWUs/Miles.
If purchasing F or C/J, I'd purchase ORD-SFO-HKG/ICN-SIN on SQ1 or SQ15.