This is a totally non-scientific answer, but ammoseek.com has a list of their most popular selling cartridges.
Today they are, in order of most popular:
1. .223/5.56
2. 7.62x39
3. .308
4. 300 Blackout
5. 6.5 Creedmoor
30-06, 300 mag and .270 come close after. My fave, the 7mm-08, is something like 16th.
