Not sure about coach, but both strive to be seen as the world's best when it comes to premium cabins, especially Suites.
Emirates is also aggressively targeting the thru traffic between Europe and Australia. They're doing a good job at it too, whereas SQ is overpriced.
All my recent trips to Sydney have been on BA in fact. Not only did I get the best price by far, but their family membership scheme allows me to scoop up the miles for all 4 of us. There's none of this Emirates/Qatar daylight robbery of giving you... 20% of what you should earn, and as for SQ, well, they have no family scheme whatsoever.
Plus my recent reservation on BA must have been some sort of special promo because not only was the price great, but for the return leg we are in Y+ at no additional cost.