I have noticed this too. I've used Apple Music mostly on cellular (T-Mobile, on an iPhone 5S), so I'm not sure if this happens over Wifi, but I suspect it doesn't. It almost sounds as if someone is jiggling a loose RCA cable on the back of the CD player, or to me, like bit-rot on a burned CD. I'm guessing it's Apple being really aggressive when it comes to dynamically adjusting things so we get a quick start to streaming and no drop-outs. If that is the case, then I'm pretty disappointed. If the music starts to distort as a result of efficiency decisions, then that's unacceptable.
As a test, I compared a song that was giving me these issues on Apple Music with the same track on Spotify (I have it set to stream at 320 kbps). The song was The Spirit of Radio by Rush. Not only did Spotify not have any distortion, it sounded noticeably better, even over a Bluetooth connection to my car stereo.
I wish Apple was more transparent about what kind of quality to expect when streaming over cellular vs wifi... but even if it was dropping to say, 128 AAC, it shouldn't be cutting out like it has been. Also, it would be nice if you could specify to only stream at 256 AAC. I would tolerate more delay with a stream for better sound any day.
I like the service so far so I hope this can be chalked up to launch-week kinks that will be ironed out in short order. But, if you're listening Apple, I'm not paying for the service no matter how much I like it unless it can stream music in an un-distorted fashion (and be used with Sonos).
Thanks!