i do believe that the RR engine has been selected by many many carriers.
BA have 16 in service with another 16 options.
Emirates, Cathay Pacific, Thai, Singapore, American, all have large orders for the 777, american has 47 on order all powered by RR-Trent-892s, Emirates must have 20 on order/in service, Singaopre like AA and UA has 40 something on order but these too are powered by Trent engines.
PW has engines on All Nippon i believe as well as UA and someone else. Just a handful of customers. GE has 30BA machines flying, and a/c being delivered to JAL??? and some US airlines such as Delta, Continental...
RR by far has the largest share in number of engines and customers.
GE has the fewest engines but maybe more customers that PW. This may change with the new -200LR and -300ER aircraft, which GE has an exclusive on.
BA may buy some of the -300er aircraft which means they will have GE again, or they may not buy the ultra-long range 300, just the boring simple -300 with trent 898s.