EK may be in front of SQ for rights across the Pacific because of a certain event in Singapore a few years ago which seriously pissed off the Australian Government.
Romulus, Apart from United, American Airlines currently operate to Oz through a codeshare with Qantas - there is no way that Qantas alone could fill six B744 services per day from Oz to LAX (plus the SFO services). Delta also has announced its intentions to operate the route.