Usage documentation for the Bugzilla WebService is entirely in the documentation of the Bugzilla::WebService modules.
The currently released versions of Bugzilla support XML-RPC, JSON-RPC and starting with version 5.0 RESTful interface. Note that in a future version, probably 6.0, XML-RPC and JSON-RPC will be considered deprecated in order to focus future development on the REST interface.
Notes on the REST interface
BzAPI is a separate service interfacing with Bugzilla and providing a REST interface, different from 5.0's native REST. It will work with Bugzilla 3.4 and up. Note that it is a completely separate application from Bugzilla; it uses existing methods (XML-RPC, CSV representations, and raw HTML) to access Bugzilla data, which it transforms into JSON and serves RESTfully over HTTP.
The Main Documentation
In many cases, you can just read this very latest documentation for the WebService, because it accurately describes the version in which every feature was added. Make sure to read the "History" section of the docs for every method that you use, though, so that you understand how it changed across versions of Bugzilla and when it was added. (You will see the "History" section at the end of every method's individual documentation.)
The "SEE ALSO" section has "Server Types", which describes specifics about how to use the WebService on XML-RPC vs JSON-RPC. It also has a sub-section called "WebService Methods", which links to the documentation for all of the actual functions you can call in the Bugzilla WebService.
If you want to read the documentation for stable versions of Bugzilla directly, here they are. In each of these pages, see the "SEE ALSO" section for links to the documentation of each function that the WebService provides.
Note also that the JSON-RPC WebService was added in 3.6, so versions before 3.6 do not have JSON-RPC support.
Versions before 3.6 don't have SEE ALSO sections, so we have to list out all the documentation explicitly:
Versions of Bugzilla before 3.0 did not have a WebService.