I only know about JAL. Untill about 8 years ago, 747 classic fleet (around 45-50 strong) was the best. It went everywhere in the network, if not by passenger, then cargo. But after 9/11 when JAL started cutting routes, 744 became the premier fleet. But within last 1.5 to 2 years, 777 has become the fleet to be.
Basically everyone fly a mix of domestic and international routes. But internationally, most pilots are grouped to one particular geographic region, such as Pacific, Europe or Asia within the fleet.