I outlined earlier that you need to convert to Asia Miles to redeem on Qatar. Asia Miles is Cathay's frequent flyer scheme, both Cathay and Qatar are oneworld members.
You'll need to:
1. Open an Asia Miles account
2. Check availability through Asia Miles for the flights you want (it will show how many miles you'll need)
3. Transfer miles from Altitude to Asia Miles
4. Redeem your miles for the flight through Asia Miles
I haven't used Asia Miles so I don't know if there are any other requirements.
Although you are right, Singapore Airlines doesn't fly one stop to Madrid, they definitely fly one stop from Milan to Melbourne as I've flown this route, so there is always the option of two one way flights - one with Qatar, one with Singapore, all depending on availability of course.