PS: My answer above was made with the thought that what the OP was requesting could not have been a Google Search Operator, because...
- none of the Google operators was really complex enough nor IMO suited to the task described unless you got very clever with them...
- there were never hundreds of them... not even close.
Though the number of Google operators was diminished several years ago, there were never very many (maybe 20 or 30 max, if even that), but we have lost a lot of free or cheap data from the search engines and from startups trying to dent the market.
Amost none of these has been called a Google Search Operator... They don't work the same, and they've been tools or suites...
- backlink tools (eg, Yahoo Site Explorer was excellent for backlinks; but the Google Link Operator was never very helpful)
- keyword suggestion tools (including Google's public suggestion tool, and also Trellian Keyword Discovery, WebPosition Keyword Gold, WordTracker, etc), perhaps good for competitive intelligence, but all really flawed with not enough data
- site comparison tools (Alexa, ComScore Compete) etc... all easily skewed by how they got their data.
Google Trends is free and better than it used to be, but it isn't very granular, sucks on anything but big mainstream phrase, and probably not useful for reverse engineering sources of clients.
Most of the keyword suggestion and traffic analysis tools, which might help you find out what people are searching for, are now bundled with subscription SEO suites, think Moz, SEMrush, Ahrefs, and SpyFu. Whenever I've looked at these or used data from them, I've seen that they have some useful data for the expensive pay per click terms, and are awful for long tail... and probably better than nothing.
Perhaps what the OP meant was some sort of utility software that might be used in search... also incorrectly described as an "operator"... to be downloaded from one of the "shareware" or "software download sites" online. I haven't really explored those in many years, but I see they are still active. Search for these phrases noted (without quotes) and variants should bring them up.... software download sites shareware
If you really are looking for Google Search Operators, as keyplyr I believe thought you were seeking, this has been the best reference I've found over time... GoogleGuide - Google Search Operators
The above is not affiliated with Google, and not all of those operators work now. Note also that the list is actually smaller than it looks, because the same operators are often repeated in several categories.
Google's own page, which I had bookmarked, has turned into pablum, and I won't even suggest it. But SEJ, or cough cough Moz might have OK blog posts. Etc... and you can use Google search, same as we have to.
Re "Google Search Operators", you wrote...
I saw the lists on Google, but they only have like about 2 dozen operators, if I remember correctly, Google used to have the list, and it was in the hundreds of operators, but maybe I'm wrong, I don't remember...maybe somebody here does?
Yes, you are wrong... maybe they had thirty at most, if even that... and IMO you owe keyplr an apology. Shower Scene was being a wise ass, but I think he was to the point. Nobody with a tool like you're asking for is going to post it free or even for a bit of money. It's up there with the "rank me first" meta tag.