If I book a trip such as JFK-SIN-MLE (New York -> Singapore -> Maldives) RT (round trip), would it be possible to use miles to just upgrade the JFK-SIN portion of the trip? Or even simpler, if I have a JFK-LAX RT ticket, can I upgrade one direction of the flight even if the return trip has no upgrade inventory?
I have verified that you can upgrade two segments of a ticket, but there has to be something special about the segments.
In this case, the trip goes JFK-MLE and stops in Singapore's hub SIN. However there is also a short stopover in Frankfurt between JFK-SIN.
After booking the JFK-MLE ticket (which has the following stops JFK-FRA-SIN-MLE) I was able to upgrade JFK-SIN, SIN-MLE, MLE-SIN, and SIN-JFK as four totally separate options.
My only guess is that if you book a ticket through an airline's hub (and this may only be the case with SQ) and you are changing planes in the hub, you can upgrade those portions separately.
This isn't a specific answer to your question since my experience is with ANA (not SQ), however, both are Star Alliance (SA) so they may have similar policies. Think of this as a warning (that was too long to fit into a comment).
My wife and I booked a round-trip IAD-NRT on ANA using miles. We then used miles to upgrade the outgoing flight but not the return flight. Once we arrived in Japan, we decided that we also wanted to upgrade the return flight after all, however, ANA refused to do so claiming that flights cannot be upgraded once the first leg of the itinerary has been flown, even if we were willing to pay cash. I believe this was due to the fact that the underlying ticket was purchased using miles; the only way to perform the upgrade was to purchase a full fare ticket and thereby lose the miles we spent for the return flight.
Assuming SQ has a similar policy to ANA (since they're both SA), I would recommend making sure that you decide beforehand which flights you want to upgrade, since you may not be able to do the upgrade later.