Starting with SonarQube 4.4, the web application offers a built-in and always-up-to-date documentation of the Web Service API. The link to access it is located in the footer of the page:
You can see it live on Nemo at http://nemo.sonarqube.org/api_documentation.
Some web services (like administration ones) are secured and require credentials that can be passed using basic HTTP authentication.
Example with curl to get the server system properties with the user admin/admin on your freshly installed SonarQube instance:
curl -u admin:admin http://localhost:9000/api/server/system
Some information documented in web services response examples depends on some plugins to be installed.
For instance :
For SonarQube < 4.4, see the archived documentation.