Is it possible to use Google to find Drupal with a specific version of Drupal? Ideally it wouldn't be possible to use Google or another search engine to find many Drupal installations with e.g. Drupal core 7.31. But I'm not sure if it's not. 

So I would like to know: is it in fact possible to find Drupal installations of a specific version?
Using Google Search it might be a little difficult for you. This search phrase can give you a few websites.

For better results you need some website profiler like BuiltWith to give you the list. The closest I came across was http://trends.builtwith.com/websitelist/Drupal-7 You have a list of Drupal 7 websites here but not 7.31 in particular. They might give you the details in pro-version but I have never tried.