I uninstalled Flash Player on my computer, now that most sites no longer need it. Adobe's Flash Player is still bundled with Chrome, so I had to disable it from the chrome://plugins/ page.
One of the only Google services that still requires Flash is Google Play Music, but only if you don't use Chrome. 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 always enabled. 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.