No, currently the specification (as of June 2015) is still in the working draft state — you may want to follow the discussion on GitHub, or get the latest spec-related highlights on Twitter.
Visit isServiceWorkerReady by Jake Archibald to track the service worker implementation status.
It depends on your use case and your target platform. Service worker support shipped in Chrome 40 stable so you certainly could use them in production, if you wanted.
If you are familiar with the notion of progressive enhancement — service workers are a perfect candidate for progressively-enhancing your website/webapp.
You should use feature detection, to check for service worker support and add offline capabilities, or notifications support for those browsers that already implement the spec.