There are some fantasy writers– I’m not going to name names and I’m not going to necessarily say, “This is bad” or such– but there are some fantasy writers who will give you lyrics and lyrics of in-world songs. Sometimes this gives you a fascinating look into history and culture. Sometimes it stops the action dead while characters sit around and sing.
Again, not saying right or wrong, it’s a stylistic choice. Sometimes it pays off.
It’s never the route I go, mostly because… not a lyricist. If I wrote songs I might have a different frame of mind. Now, I do have a poem in A Murder of Mages, but it’s plot-relevant. Songs typically aren’t. They are a typically a pleasant diversion at best, a nice bit of atmosphere to set tone and enhance worldbuilding.
Because that’s the important thing: music and songs are a critical aspect of culture, and thus part of worldbuilding. Just ignoring it doesn’t work any more than having it freeze up the narrative.
For example, in my backburnered work-in-progress Banshee, Lt. Kengle is the only human on a ship of aliens. The only thing she really has to connect to, emotionally, is her music. Her music is, of course, 24th century music, and I really don’t go into details about what, exactly it sounds like– though I do highlight her preference for female singers (and that it is, still sung). What I do go into is how listening to her music makes her feel.
Would you be able to hear it, reading the book? Honestly, probably not. But what you would get out of it is what it means to Lt. Kengle, culturally and personally. That same technique can be applied– in fantasy or sci-fi– to any pop culture element: music, drama, sports, lifestyle. The msot important aspect of it, in terms of your story, is not what it is, but how your characters feel about it.