Well - do you mean popular in Europe or popular European calibers?
Probably the #1 in use or becoming it is the .308. In Finlabd, it holds an about 70% market share.

As for European, in the Nordic countries the 6,5x55 is very popular.
I see much less 9,3x74R. Much more common - esp. for bear - is the 9,3x62.
I have only ever seen four 8x57 in use. It's making a little comeback, I think, with bear and wild boar increasing in numbers. It's a good, effective, mild-recoil medium (or even heavy).
The 7,62x53R is becoming rarer. Of course, no new guns are made for it save the Baikal single shot. Pity, it'd be excellent in break-open guns.
In Central Europe I believe you see a lot of 7x64 and 7x65R, not so in Finland. I know of two including mine.

To see bigger than 9,3x62 is rare and there the top gun is the .375 H&H.

For smaller, the .222 is popular. Many guns are stamped 5,7x43 instead.
But I believe the .223 is becoming more common....

- Lars/Finland