Martin Splitt from Google was asked on Reddit "are there plans to support service workers?" This would be specific to Google Search supporting it. Martin responded that no, Google has no such plans. He then went on to explain why Google has no plans on supporting service workers.
Martin said the reason why is because Google Search does not want to show different searchers different content or information for the same search result.
Martin Splitt said "as we have to assume that someone clicking on your page from a SERP is a first-time visitor, running a service worker is usually not going to do much good - because Googlebot would then likely see a different experience of some sort than a first time visitor would, which isn't going to be great for those coming from a SERP that promises something different than what comes back."
So I guess don't bet on this happening any time soon...
Forum discussion at Reddit.
Update in 2023: Google's John Mueller said this has not changed in 2023, he posted on Mastodon, "I don't think anything has changed. I wouldn't expect it to change - it's computationally expensive to run service-workers in the background like this for indexing."
