This is actually my most hated bug in this game. What can I say? I really like the music in this game. And many people are not affected by this bug since it only occurs if you make certain choices. So, it seems to have gotten overlooked. I will start with the short version.
TLDR
Sera got mad at you and stole the minstrel's songs.
FULL EXPLANATION
Now, I have played exclusively on PC, and have the full version with all the DLCs, but I have seen the occasional post regarding this for both Xbox and PS users as well so I suspect but do not know that this issue may affect all versions.
The problem begins in Haven. You may have noticed that if you leave the tavern and return, or simply wait for songs to finish, you get collection notifications for each new song. This is great until she runs out of new material and STOPS SINGING! But don't panic quite yet. This story isn't done yet.
Once you meet Sera in a dark alley you have the chance to have her join the party. If you invite her to join, the minstrel learns a new song, and will now continue to play normally again. If you meet with Sera but choose not to invite her, the minstrel will NEVER SING AGAIN! At this point, the only way to fix this is to load an earlier save or start a new game, and make different choices.
The last part of this bug is Skyhold Fortress. If Sera joins before or even possibly after going to Skyhold, everything should be fine. Once you have the minstrel playing normally in the Skyhold tavern you can relax. If you are so inclined you can now ask Sera to leave, or not, whatever you like.
Since the last patch for DAI was back in 2015 I seriously doubt anybody is going to get around to patching this one, but this should at least give you enough information to keep the minstrel working normally in your own play through.