Hi. I've been looking for Singapore tickets for early 2012, and in my serious searching, I've discovered that for "value" the two best airlines are Singapore Air and Cathay Pacific. Both offer flights from LA and San Francisco. Singapore Air has "direct" flights...meaning you stop briefly (presumably for refueling), but you do not change planes. Cathay has a plane change and a longer layover. When you are flying that far (and from the US it is a LOOOOOOONNNNNNNGGGGG flight), I put a premium on faster and no possibility (or greatly diminished possibility) of lost luggage and missed connections.
Price between the two was pretty much identical (within 20-50 bucks...not huge). Best prices are flights Tues/Wed/Thurs.
Absolute cheapest (per Orbitz) was Phillipines Air, but longer travel times by quite a bit. Savings was in the order of $200. However, in terms of travel quality/service etc there simply is no comparing Singapore Air and Cathay. They consistently rate as the best airlines in the world. You will be blown away by the service difference.
Cathay does offer a non-stop from Chicago to Hong Kong, and from there a fairly short hop to Singapore. However, the layover was 10+ hours, making the total travel time 12 hours longer than other options. No thank you. LOL!
Also, I curiously discovered that booking the connecting flight to the coast (I live in Minnesota) in order to catch a San Francisco flight was best done separately. For whatever reason, both airlines default to offering you FIRST CLASS tickets from Minneapolis to the coast, with no option to say "no economy would be fine, really

). So, I'm planning on spending a couple of days in San Francisco on both sides of the trip (to allow for cancellations, etc).
If you book with Cathay or Singapore, the cheapest price was from their website.