Here is a list from 2006 of the "fastest growing" routes between 1996-2006. I assume these are probably the busiest long haul routes and summarily would have the most movements, although the A380 may have changed that:
1. London-Dubai up 848,270 seats
2. London-Chicago up 552,691 seats
3. London-Hong Kong up 481,675 seats
4. Melbourne-Singapore up 470,023 seats
5. Sydney-Singapore up 441,954 seats
6. London-Singapore up 407,773 seats
7. London-Mumbai up 390,940 seats
8. Dubai-Singapore up 370,630 seats
9. London-New York up 369,885 seats
10. Brisbane-Singapore up 308,952 seats
London-Dubai and London-Hong Kong both appear to have ~84 weekly flights (12/day).