Which aircraft type flies the most annual passenger-miles?
Some heavies (747, 777, A380) fly short- to medium-haul in Asia and the Middle East. You could say they're stealing jobs from 737's and A320's.

Anyway, what I could find was for the US only.

Checking the Revenue Passenger Miles Total by Aircraft Type for 2016 on transtats.bts.gov we find—


All 737 variants: 153.2 billion
All 777 variants: 38.3 billion


Expanded:

Boeing 737-100/200         18,374,213
Boeing 737-200C                58,536
Boeing 737-300          6,609,070,471
Boeing 737-300LR              574,860
Boeing 737-400          1,484,593,783
Boeing 737-500            324,819,083
Boeing 737-700         44,485,736,725
Boeing 737-800         69,014,076,197
Boeing 737-900         18,203,949,655
Boeing 737-900ER       13,058,102,746
Boeing 777-200-        32,808,990,997
Boeing 777-300-         5,466,846,693


Top types with at least 10 billion pax miles:

Boeing 737-800                  69,014,076,197
Airbus Industrie A320-100/200   48,907,277,978
Boeing 737-700/700LR            44,485,736,725
Boeing 777-200ER/200LR/233LR    32,808,990,997
Airbus Industrie A321           28,492,821,887
Boeing 757-200                  26,182,350,221
Boeing 767-300/300ER            24,176,619,545
Airbus Industrie A319           22,087,734,671
Boeing 737-900                  18,203,949,655
DC9 Super 80/MD81/82/83/88      15,379,739,818
Boeing 737-900ER                13,058,102,746
Airbus Industrie A330-200       10,556,831,804


From this top list:


Single-aisle: 285.8 billion
Widebody: 67.5 billion


(Narrow- vs. widebody above is just in case you are wondering. Arbitrarily selecting top-x won't reveal much.)



Boeing's outlook for the future based on past- and current-trends sees the single-aisle market outgrowing the widebody.







I've tried to dig deeper, but it seems some 260 governments don't openly share those statistics, we'd need to hire an analyst firm for a worldwide view.