Singapore and United are both part of the Star Alliance so you can use Singapore KrisFlyer Miles to book your United flight.
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You can also use KrisFlyer miles to enjoy award tickets to more places with Star Alliance* airlines. Award redemptions can be done on singaporeair.com and will be implemented to all Star Alliance partners progressively.
You can join multiple member airline frequent flyer programmes if you wish, but they are each managed separately, so you cannot transfer miles or points between them or combine them into one. By using a single frequent flyer programme whenever you fly, you will reach Star Alliance Silver or Gold status faster.
As a founding member of Star Alliance, Lufthansa offers innumerable advantages in the world's biggest global network. Its regional strategy under the Lufthansa Regional brand connects Europe's regions with one another and with the world beyond. Bilateral partnerships augment and enhance the services portfolio.
Members of the Star Alliance include: Aegean Airlines, Air Canada, Air China, Air India, Air New Zealand, All Nippon Airways, Asiana Airlines, Austrian Airlines, Avianca, Brussels Airlines, Copa Airlines, Croatia Airlines, Egyptair, Ethiopian Airlines, Eva Air, Lot Polish Airlines, Lufthansa, Scandinavian Airlines, ...
It's not illegal to buy KrisFlyer miles. There's a lot of things in your browsing history that will make the FBI kick down your door, but buying KrisFlyer miles is not one of them. That said, the selling of KrisFlyer miles and award tickets is against the T&Cs of the KrisFlyer programme.