I think with UA now out of that market [SIN], fares will not soon again be as low as the $500 ex-ORD, etc. we saw a few months ago on their web site. Capacity is being reduced significantly over the Pacific, so the only thing to keep fares low will be a steep fall off of tourist traffic to SEAsia. The business market has already tanked.
But the tourism market may also be soft this winter. Afterall, the UA fare that was sold on the web was actually their inhouse consolidator fare which is used along with land packages. Low sales of such vacation packages left UA with lots of unsold seats in this fare category. That's why it was restricted to web bookings.
For the time being, as suggested, look into weekend web specials.