Android Central

Google previously added support for YouTube videos to play at 60fps on the desktop site, and later added that same frame rate for live streaming videos. Now the official YouTube Android app can handle 60fps playback, if the video already supports that high of a frame rate, and the mobile device is capable of handing it.

The new options show up when you begin playing a video and then tap on the screen. You should see a menu icon on the upper right side, which you can then tap to bring up some options. It's the "Quality" selection that you want to then tap on to bring up a menu to see the video quality options. If the video supports it, you should now see "720p60" and "1080p60" among the choices you can pick from.