Our Getting Started guide gets you up, running and debugging a "Hello world" Service Worker in Chrome. If you are stuck the FAQ may have the answer you need.
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!