I've found the practice of looking at search-term frequency, across a domain or set of domains (using the "site"" Google search filter) to be useful, for example the "Top 100 Global Thinkers" report linked below.
It uses 100 search terms -- "global thinkers" identified by Foreign Policy magazine -- searched across a set of about 100 domains and TLDs, largely social media, various journalism (newspaper / magazine), and a few institutional sites, as well as selected national and other top-level domains. The result is an interesting profile of where more robust online discussion or commentary might be found.
https://www.reddit.com/r/dredmorbius/comments/3hp41w/trackin...
The full report requires running roughly 100 x 100, or 10,000, Google searches. I'm finding that it's necessary to space these ~5-10 minutes apart, which means that the full analysis takes over a month of wall-clock time, from a single IP.
I've considered several possible follow-ups to this study, including more or alternate domains, different keywords, and various other variants, but both the run-time and codeing to bypass bot-detection put me off this.
I've tried reaching out to Googlers I know to see if there's any possible alternative means of acquiring this information, to no avail. I've also looked for various research interfaces or APIs, with no joy.
DuckDuckGo and other search sites don't have the rate-limiting (I've used them for other purposes), but also don't have the (granted, often very inaccurate / imprecise) match-counts which Google offers.
Putting this out there both as an example and a request for suggestions as to how I might improve or modify the process.