The poem Oliphaunt is in the Lord of the Rings story recited by Sam Gamgee in The Two Towers (Book IV, Chapter 3), the volume where it was first published. Sam Gamgee explains "That's a rhyme we have in the Shire [...] we have our tales too, and news out of the South, you know." This bestiary poem is related to an earlier one imagined by Tolkien called Iumbo.
The Adventures of Tom Bombadil and Other Verses from the Red Book contains 16 poems, each with their own plot. It was first published in 1962. I have read a 2014 edition of the book, edited by Christina Scull and Wayne G. Hammond. It contains extra content, such as earlier versions of the poems, and commentary.