How can one figure out what version of Bitcoin-Core is used most? Where can one find statistics on how many nodes are running older Bitcoin Core versions or alternative clients?
Yes.

Go to https://getaddr.bitnodes.io/nodes/, click on the "User Agents" tab.

At time of writing:

1   /Satoshi:0.11.0/    2119 (35.22%)
2   /Satoshi:0.10.2/    1416 (23.53%)
3   /Satoshi:0.10.0/    437 (7.26%)
4   /Satoshi:0.10.1/    418 (6.95%)
5   /Satoshi:0.9.3/     352 (5.85%)


(You can also click on "more.")