Listening to Apple Music on iPod Classic

You cannot listen to Apple Music on the iPod Classic. Apple Music is a subscription service and therefore requires an internet connection in order to ensure that the subscription is still active.

Your options are to buy the music and download it into your iTunes Library, for syncing with the iPod Classic, or use an iPhone.

I'm not an Apple Music user, so I'm not 100% sure how it works, but I assume that if you use Apple Music on an iPod Touch, then without a Wi-Fi signal, it cannot confirm the subscription and so music would not play.

... and of course, with an iPod Touch, you are not guaranteed a permanent Wi-Fi connection.

Apple Music items cannot be transferred to an iPod classic.

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This page confirms that you can have Apple Music on an iPod Touch

Join Apple Music on your iPhone, iPad, iPod touch, Mac, or PC - Apple Support

I imagine it expects to check periodically, so if you were out of range of wifi for long enough it might stop working until you got back into range.