SecretFlying.com has a link to a $477 ticket for LAX to Hong Kong (United), Hong Kong to Singapore (JetStar Asia) several days later, then Singapore to LAX on United.
The real deal here is LAX to Singapore is currently $694 round trip on United.com, while LAX to Hong Kong is as low as $433 round trip.
These are for departure dates from late August into November.
Attempting to replicate the itinerary on United.com did not work, but there are definitely lots of possibilities for flying LAX to Singapore with a 22 hour layover in Honolulu, a transit stop in Guam and and overnight in Hong Kong for $530 round trip. You can fly LAX-Singapore round trip with overnights in Honolulu and Hong Kong for $164 less than a simple LAX-Singapore round trip on United.
Another interesting option is flying an open-jaw with Singapore and Hong Kong via Honolulu and Guam or flying from Hong Kong via Guam and Honolulu. I failed to find a low fare allowing an overnight transit layover in Guam, but you can book a 22 hour overnight in Honolulu flying LAX to Hong Kong via HNL and Guam or a 20 hour layover in Honolulu flying HKG to LAX via Guam for $519 to $533.
One way flights from Hong Kong to Singapore are as low as $75.
Mon Sep 9 LAX-Honolulu 21hr 30min overnight. Arrive HNL 4:55pm Mon and leave HNL 2:25pm Tue Sep 10 for Guam.
Arrive Guam 6:05pm Sep 11, depart Guam 6:55pm
Arrive Hong Kong 9:50pm Sep 11, 18hr 50min overnight layover,
Depart Hong Kong Cathay Pacific 4:40pm Wed Sep 12
Arrive Singapore 8:35pm Wed Sep 12
Depart Singapore Mon Sep 23 8:45am – Arrive SFO Sep 23 8:50am
Depart SFO Mon Sep 23 11:00am, Arrive LAX 12:42pm
United Airlines $459.55 round trip LAX-HKG via Honolulu and Guam with overnight in Honolulu each direction
Monday Nov 11 LAX to Honolulu 22hr 22min layover
Tuesday Nov 12 HNL to Guam arrive GUM 7:05pm Wed Nov 13 (date line change)
Wed Nov 13 Guam to Hong Kong, depart 7:55pm, arrive 11:00pm Nov 13
Mon Nov 25 Hong Kong to Guam to Honolulu 15hr 23min overnight layover
Tue Nov 26 Honolulu to Los Angeles arrive 5:25pm
United $518.23 Open Jaw LAX to Hong Kong/Singapore to LAX
There are all kinds of itinerary possibilities with a chance to visit several places across the Pacific for $460 to $600 round trip.