OK, so Joe and I looked into this for over 2 hours and I think I have figured out what might be happening. Follow these instructions. This is happening with only Playlists (at least that is what has been reported) but this should fix it for all instances.
If you have the rights to the music, be sure each song has the Display Embed Code turned on.
Not sure if this is something that NEEDS to be done, but it is worth doing to be sure.
Be sure you have the Playlist is set to Public and not Private. Now, even if your playlist is set to public, click on private and then public again, it seems to reset it to the public mode then click on Save Changes.
DO NOT use the URL from the playlist window, instead, click on the share button and you will see this window appear:
Use this URL for the stack setting.
After doing all three of these on all of my playlists (many of which would NOT display) now display on my test site here
If you follow all of these instructions, then your playlists should reappear.
Let me know if it does.