As a Star Alliance airline, your United Mileage Plus account will be just fine to collect your Singapore miles. Just present that when checking in.

Your British Airways flights both would have counted into your AAdvantage account, too - massively! They're both in the oneworld airline alliance. If one was relatively recent, it's worthwhile trying to claim (if you still have the boarding pass).

You can't merge the two: oneworld isn't Star Alliance. It would have been better to have stuck to one airline alliance for all your flight: in this case, flying ORD-LAX with United, US Airways or Continental (who are all Star Alliance).

Ignore the airline brand: it's the alrline alliances that do the hard work.