Hi AmeriBrit, welcome to flyertalk.

Personally I'd choose BA as they have a WTP cabin, SQ don't.

However, if you'll be based in Singapore, your only options to earn BA miles are
* Australia (for now, Qantas is reducing their presence in singapore)
* London (BA)
* Hong Kong (CX)
* Tokyo (JL)
* KL (MH)

Joining Kris Flyer will allow a lot more direct flights.

For your current travel plans of London and the U.S.

London: BA fly, and have just started (or are starting) using the new 777s to

Singapore. SQ have an A380 I believe. They also fly to Manchester.
America: I believe there are few direct flights from Singapore to the states, and the ones there are are very long. You'll most likely want to change in Hong Kong or Tokyo, which means you'll be flying Cathay or JAL, both will earn you status on BA.

3 trip to the UK will get you 600tp (in WTP), enough to maintain BA Silver, the trips to the states, if on CX, will add to that. You could probably try SQ out once and see how it goes.

If you're not going to be doing a lot of short flights around the region, I'd personally stick with BA, and join Agean for the occasional flight on SQ. If you did one London trip on SQ in business class, I think you'll be close to getting Star Alliance Gold with Agean, which gives you lounge access when travelling economy.