When redeeming DL miles, any combination of SkyTeam flights is permitted. But SQ can only be combined with DL on the domestic legs and cannot be combined with any other airlines at all. For example, you cannot fly DL ATL-FRA and SQ FRA-SIN. The award must be DL from home to gateway (if not already in the gateway), SQ from gateway to destination.

To the OP, you can probably book an open-jaw itinerary, such as KE USA-ICN-BKK and KE SIN-ICN-USA then buy a separate one-way ticket on another airline to close the open jaw. There are lots of flights on lots of airlines between HKG and BKK and BKK and SIN, plus heavy competition on SIN-HKG.