I don't feel very qualified to answer this because I've only flown with a smaal percentage of all the airlines that have existed, and only fly economy as I have to pay for myself but I'll have a try from the ones I have flown with.
On Board Service In All Classes, Catering, Seating etc,
on Transatlantic routes
2. United Airlines
3. British Airways
4. Singapore Airlines
5. Virgin Atlantic
Customer Care and Service
1. United Airlines
2. British Airways
4. Virgin Atlantic
5. Britannia (amazing for a predominantly charter airline)
Fleet Diversity and flexibility
Don't know enough about this. I guess all the major airlines have this to a degree. Without wanting to start A vs B, I tend to like to see big Boeings and small Airbuses.
Star Alliance (Not an airline, I know, but look where its members will take you!)
Just my 2p worth