Poetry in Works of Fiction

I'm not speaking here of rhythm, rhyme, alliteration, etc. in prose, but rather actual undisguised poems.  Should they appear in fiction -- fiction being written for the current market?  "Of course they should," I say, but how and to what purpose?

Think of Lord Of The Rings (I realize this is anything but current), and what I think of as its three kinds of poetry:
• poems from the old days (in English)
• poems from common folklore (in English)
• poems from the elves (in Elvish)

The first are long "chants", and (I regret to say) too often tend to the boring.  A good example is The Song of Nimrodel, which is so long Legolas doesn't even remember it all.  You can barely call these backstory.  It doesn't matter -- they're too long to appear in that form in a contemporary work.  Most readers simply won't stand for it.  I didn't skip them, but when I saw the poem went on for a page or more and the first stanza proved the entire poem to be of this ancient ilk, I skimmed.  I admit it.  Some of these poems are beautiful, and when shorter (1 - 4 stanzas) are eminently readable.  Poems demand a different style of reading than normal prose.  Only sometimes could I be persuaded to break my normal stride of reading to slow down for these longer old-style poems and "speak the lines inside my head."

The second are phrased in simpler language, and tend to be shorter.  Often much shorter.  Example: The Road Goes Ever On.  Sometimes they're songs of which we're given just a verse or two.  That's fine -- they set a mood, give us a flavor of the time / world we're moving through.  I'd slip into and through these poems easily without coming out of the text, without breaking the spell of the story.

The third are (literally) unreadable.  Example:  Namárië. You could pour over them if you wish, but if you're reading rather than studying, the elvish is impenetrable.  I could have used a translation, I suppose, even if it didn't capture the rhythm or rhyme (if any).

To come back to the main issue of this entry, the question is what place, if any, poetry has in a modern work of fiction.  Not every work deserves poetry, or has a place for it, but some do.  Some stories beg for children's songs and taunts, mnemonic jingles, spells, etc.  But I think they should be used as an accent, rather than be an overpowering influence.  To that end they should be kept short.

My first novel has a few of them, all short, and all in the form of nursery rhymes.  I think they fit and it makes sense to have them there.

I'm behind on my reading of current fiction (I'm trying to do better) -- do you know of any recent examples, either egregious or salutary?