Will be interesting to see if it can be done... and I'm keen to know!
The yes it can be done case: SQ can check someone into a VA metal flight, although I've only seen it done on a SQ codeshare, SQ ticketed onto a VA flight. (Friend was flying SIN-BNE-CNS and as a status-less pleb was put in 3A BNE-CNS by SQ, sending me a pic of her boarding pass with a smile. I had my hangover friendly 3C SYD-CNS that day too) The SQ System (see later) has to talk to VA's system so I am inclined to say that if you have your Ticket Number (not just your PNR) it should be possible, the unknown is the check in counter staff will do it in SIN.
If the Check in counter says no - try the transfer desk in SIN as they are awesome and the lounge staff are a bit useless.
The case against: SQ ground staff can be really hit and miss. Someone in another thread mentioned that its possible to check in using Amadeus, and that's true (TG offered to do it for me with my connecting SQ flight) however SQ run Amadeus where as VA run on Sabre and that might just break things when not ticketed as SQ.
In future, why not book through a travel agent and then you can book both flights on the one ticket number (something you're not able to do yourself) meaning you could be checked all the way through. Also you are what is called an end-on-end ticketing. One ticked to one location and a second ticket to another location, meaning you have no protection for missed connections
Even if its $20 more its less than the VOA!