Clearly left-click is the most common, but how often are others like right-click and double-click used? What is the distribution frequency of those inputs? What is the expected result of those mouse inputs in web based applications?
As close as a desktop application. Web Applications in general, and Cloud based document processing in particular try to mimic the behavior of desktop applications as much as possible. A right click on the marked text brings up the options menu, and double click typically opens the document in a document processor. Just the same behavior as a desktop experience.
In our tests the number [that double click] tends to be around 10% of the test subjects, typically aged 50+. Furthermore, there appears to be a high correlation between the users who double-click and those who are generally “insecure” web users - source
Double click is used to open documents and such from your desktop. The web handles single- and right clicks differently, but still people want to open things by double clicking it. This can have unwanted results like adding two items to a e-commerce shopping bag. For that reason I would handle double clicks the same as single clicks.
What expectations are for right clicking isn't really documented as far as I know. We all know it brings up an options tab. On the web these are normally browser options like previous, reload or show page source. From a personal point of view, I expect to see these options when right clicking something on the web. I usually use it to inspect an element in Chrome for developing purposes or to save an image. But I'm pleasantly surprised when custom options show up containing stuff that makes it easier for me to use the web application.
So in short: treat single and double clicks the same for people expect it to open something. If a custom options tab improves usability of your application, add it, please do, but it differs from what the convention is: the standard browser option tab.
Additionally, depending on the application, if there's large wads of text, many users double-click to highlight words as they read.