UA and SQ do not have an anti-trust immunized and revenue sharing JV (like they do with LH, AC, and some other carriers) and don't actually coordinate on fare filings. The higher fares OP is seeing on UA is due to the Flight Application section in the fare rules of the UA fares. The $3K-plus roundtrip fares (like VH100NCE/TPIN fare below) are currently the cheapest SFO-MAA UA fares that permit SQ metal.

This is not actually an issue with SQ not providing inventory on their flights to UA as some have speculated. Note that these are entirely UA fares (which cover the entire SFO-MAA itin - they are not hop-by-hop). SQ is not providing access to their fares as no SQ fares are involved in these bookings. They are only providing inventory availability on their flights to book UA fares on them.

As noted in a prior post, the cheaper UA fares on SFO-SIN cannot be combined with cheap SIN-MAA fares on SQ on a single ticket due to fare combination rule restrictions in the fare rules. There is no point in attempting to do this. They must be booked on separate tickets.