I guess it depends on the fare rules and the interline ticketing agreement between UA and SQ. For example, if you flew a UA sector SIN-HKG and then HKG-SFO on SQ on an SQ C fare then on UA that fare may also book into C, but it may also book into D (discount business) on UA depending on the agreement the two have. This instance you would probably be ok because there is F/C/Y service on the route.

When you are domestic US on 2-cabin F/Y routes, the C fare may not allow F (or A) to be booked. It may say something like "If no C, book Y" etcetc. I believe SQ also used to do this on their own services, like for example when KUL-SIN and vv was F/Y service only, I believe C fares booked into Y, not F.

Now, if you found a UA flight with C being sold, then that MAY be allowed on your particular SQ C fare, it depends.