The website will generally show you what’s available and is pretty good at getting all the options displayed. However, if you can link different available sectors together for what you believe is a valid routing, you can call SQ and do it over the phone. If you ask, they will almost always honour the 15% discount, so long as what you’re trying to do is not possible online. However the way SQ do their award availability is strange, and sometimes you come up against the problem of “quota”. All SQ awards availability is done from origin, irrespective of destination. This means if you fly, say, HKG-SIN-SYD, you need to find a day when that routing appears on the website. Finding the two segments (in this case HKG-SIN, then SIN-SYD) together is often not enough, even when they’re completely different flights, flight numbers and aircraft. You can ask them to send a request to the station in question to release seats, but reports of the success of this are very mixed, even when the component parts are all available.
Remember, you can still get a stopover. All return award tickets are entitled to at least one for free. Saver level one ways are not, but you can pay (I think) US$100 for one. This, in theory, will also get around the “quota” problem above - you should be able to marry segments on different days no problem. However, I’ve had problems even with a stopover, there doesn’t seem to be rhyme or reason to this “quota” thing, and nobody at SQ can ever explain it clearly.