Our Getting Started guide gets you up, running and debugging a "Hello world" Service Worker in Chrome.
Working with a service worker is a little different than normal debugging. Check out Service Worker Debugging for details.
Service Worker shipped in Chrome 40.
Is Service Worker Ready? tracks the implementation status of Service Worker at a fine-grained, feature-by-feature level in many popular browsers.
Go to http://crbug.com/new and include "Service Worker" in the summary. You should get a response from an engineer in about one week.
If the browser crashed while you were doing Service Worker development, go to chrome://crashes and cut and paste the Crash ID into the bug report. This kind of bug report is very valuable to us. (If you do not see crashes in chrome://crashes it may be because you chose not to send usage statistics and crash reports to Google. You can change this option in chrome://settings .)
To give feedback about Service Worker that is not specific to Chromium, use W3C's public-webapps mailing list. Use the spec issue tracker to file bugs against the Service Worker spec.
For important announcements and technical discussion about Service Worker, join us at firstname.lastname@example.org.
...interested web developers are welcome to poke around too!