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Hey! Welcome to the community :)
How much free storage space do you have on your device?
Spotify won't start synching normally unless you have 500MB-1GB of free space.
Wifi - but I figured it out - although not clear in the instructions -
From the Playlist screen, click Edit then click the circled arrow on the right of the playlist you want to download. Then they started syncing. Thanks for the follow up though.
After further investigation, I determined that Spotify IS downloading songs offline. I just did not understand how to read the icons.
I've been testing Apple Music and Spotify across my apple environment (iphone 6, ipad, mac). It seems there are too many little niggling problems with Spotify, whereas Apple Music "just works". Apple has a much more attractive family plan, so I will give my money to Apple.
I have about 3.5gb left and my local files are worth 1.8gb yet songs won't transfer to the phone. I can see the songs but can't play it on the phone. They used to play before but by mistake I removed download button and when I again try it, songs won't transfer
Ok after a few weeks of having the issue on my iPhone where my playlist would not show the green downward arrow as a sign that the songs were fully downloaded and also the indicatioun saying waiting to download being present I finally figured out a way to resolve. I tried everything from reinstaling app, deleting playlist, etc etc. Here is what fixed it for me.
I went on my desktop and created a playlist called TEST. I copied all the songs (266) on te other playlist with the issue to the one called TEST. I then completely deleted the original playlist with the issue. I then went to my phone and selected to download the new playlist called TEST which to my surprise only had (263) songs that downloaded. I then renamed that playlist what the other playlist was originally called. It seems that there were three songs that were causing the issue however it would show that all songs were downloaded on the original playlist and none were on green, perhaps these songs were removed by Spotify or something, either way I now have a fully downloaded playlist. I hope this helps someone and saves a lot of headaches as it did for me.