LAX-SIN, as posted earlier, is the longest current route at 7621nm, taking about 18 hours. It's being operated with an A340-500 (The Singapore Airlines A345 Leadership). The plane has been customized with only 117 Executive Economy seats (20" wide, 37" pitch) and 64 Raffles Class seats (SpaceBed seats w/64" pitch), so there are far fewer seats than would normally be on an A340. This weight reduction allows them to achieve that range.
Out of curiosity, I plugged some dates in April on this route. For a 1 week roundtrip LAX-SIN:
Econ - $1121 or $.14/mile
Raffles - $4861 or $1.56/mile
That's pretty competitive, considering what some East coast-Europe routes cost for similar seating on similar carriers, and those routes as far.
SQ will introduce JFK-SIN soon with the same equipment, at which point it will become the longest route at 8287nm. OUCH.