One of the only Google services that still requires Flash is Google Play Music, but only if you don't use Chrome or Internet Explorer 11. I tried opening Google Play Music in Firefox and Safari and got this message: "Missing Flash Player. You need the latest Adobe Flash Player to listen to music."
There's actually a Lab experiment called "HTML5 audio" in the settings and it's grayed out and disabled. It claims that it allows you to "listen to your music without the need for Flash. Support for this lab is still experimental, and it may not work in all cases," informs the description.
Well, it looks like the Lab experiment only works in Chrome, not in other browsers. It's hard to tell why HTML5 audio is still an experiment and not a regular feature that works in all browsers. YouTube's HTML5 player works well in most browsers and Google Play Music uses the same back-end.