The line, and rhyme:

Rhyme, scorned by some major English poets (such as Milton in his epics), and rarely used in classical poetry (though rhyme re-entered Latin in the late Rome period and early middle ages), is intimately tied to the line in English poetry, even when it's internal, as it so often is in Auden,  rather than occurring at a line's end.  As a means for everything from pleasing sounds to humor and irony to managing tempo, rhyme is indispensable for most lyric poetry in meter and is often used for long poems, as in the rime royal used in The Rape of Lucrece by Shakespeare:

     O happiness enjoyed but of a few!
     And, if possessed, as soon decayed and done
     As is the morning silver-melting dew
     Against the golden splendour of the sun!
     An expired date, cancelled ere well begun:
       Honour and beauty, in the owner's arms,
       Are weakly fortressed from a world of harms.
-
     Beauty itself doth of itself persuade
     The eyes of men without an orator;
     What needeth then apology be made,
     To set forth that which is so singular?
     Or why is Collatine the publisher
       Of that rich jewel he should keep unknown
       From thievish ears, because it is his own?

Easy generalities disintegrate confronted by the body of work in rhyming English, including the endlessly repeated notion that English is a rhyme-poor language.  English is not a rhyme-poor language, but one that requires a substantial variation of the rhyming technique one might find in Italian or French.  How do you think Irving Berlin wrote all those songs -- by inventing words?  He understood rhyme differently than he would had he written in French.   There are tens of thousands of exact rhymes in Romance languages.  There are very few in English.  But there are tens of thousands of other kinds of rhymes in English, including the many perfect and near rhymes used by Shakespeare in Lucrece...

While it is true that rhyming verse includes most doggerel, nursery rhymes, verse fairy tales, and occasional verse, most as unnoted for their contributions to the art of poetry as the hackwork made seriously over the centuries, it also includes most of Dickinson and Burns and substantial portions of most major poets from 1500 to 1900, and not a few since.  Rhyme is invaluable in songwriting, from the strong lyrics of a rock song to the spirited wit and irony of Stephen Sondheim or Cole Porter. Rhyme is used by poets who use metrical prosody and by those who don't.  Many Beat poets used rhyme extensively, though they almost never used meter.

There are a limited variety of rhymes:  perfect or exact, near, assonant, consonant, and falling rhyme.  Several others, sometimes subsumed under the latter two, include slant and distant or off rhyme.

Perfect rhyme:  where the pattern repeats, the sound is the same on the last syllable or combination of syllables

           A Robin Red breast in a Cage
            Puts all Heaven in a Rage.

                    from Auguries of Innocence, William Blake (1803)

         Childhood was cloth'd in white and green to show
          His spring was intermixed with some snow:
          Upon his head nature a garland set
          Of Primrose, Daisy and the Violet.

                    from Of the Four Ages of Man, Anne Bradstreet (1650)
 

     So now I have confessed that he is thine,
     And I my self am mortgaged to thy will,
     My self I'll forfeit, so that other mine,
     Thou wilt restore to be my comfort still:
     But thou wilt not, nor he will not be free,
     For thou art covetous, and he is kind,
     He learned but surety-like to write for me,
     Under that bond that him as fast doth bind.
     The statute of thy beauty thou wilt take,
     Thou usurer that put'st forth all to use,
     And sue a friend, came debtor for my sake,
     So him I lose through my unkind abuse.
       Him have I lost, thou hast both him and me,
       He pays the whole, and yet am I not free.

I sprang to the stirrup, and Joris, and he;
I galloped, Dirck galloped, we galloped all three;
"Good speed!" cried the watch, as the gate-bolts undrew;
"Speed!" echoed the wall to us galloping through;
Behind shut the postern, the lights sank to rest,
And into the midnight we galloped abreast.

Near rhyme:  Where the rhyme repeats, the sound is close but not exactly the same (given the varieties of dialect, this is probably typical of even most "perfect" rhymes in English).

                     Will't please you sit and look at her? I said
                    "Fra Pandolf" by design, for never read
                     Strangers like you that pictured countenance,
                     The depth and passion of its earnest glance,
                     But to myself they turned (since none puts by
                     The curtain I have drawn for you, but I)...

                            from "My Last Duchess," Robert Browning (1842)
 

The first two lines of the clip are perfect rhymes.  The second pair are near.  The third pair are nearer.

Friend, your fugue taxes the finger
  Learning it once, who would lose it?
Yet all the while a misgiving will linger,
  Truth's golden o'er us although we refuse it-
Nature, thro' cobwebs we string her.

          "Linger" and "string her" are a near rhyme.  "Lose it" and "refuse it" might be in some dialects.

                      So we'll go no more a roving
                      So late into the night,
                      Though the heart be still as loving,
                      And the moon be still as bright....

                                from "So We'll go no more a roving," Lord Byron, 1817

"Roving" and "loving," in 1817, may have been exact rhyme but we don't have any direct means of finding out.   Now, outside of Liverpool, they're near, though some may argue strictly for their being consonant rhyme.
 

        She'd the brooch I had bought
                      And the necklace and sash on,
              And her heart, as I thought,
                        Was alive to my passion;
      And she'd done up her hair in the style that the Empress
          had brought into fashion.

"Sash on" and "passion" are a fine near rhyme.
 

 Cold-blooded, smooth-faced, placid miscreant!
    Dabbling its sleek young hands in Erin's gore,
  And thus for wider carnage taught to pant,
    Transferr'd to gorge upon a sister shore...

               from "Don Juan," Byron (1824)
 

Consonance:  In consonance, when a rhyme repeats, only the original consonant sound is heard.
 

     And think, this heart, all evil shed away,
         A pulse in the eternal mind, no less
           Gives somewhere back the thoughts by England given;
       Her sights and sounds; dreams happy as her day;
         And laughter, learnt of friends; and gentleness,
      In hearts at peace, under an English heaven.

"Given" and "heaven" are a fine consonant rhyme.
 

          This is my letter to the world,
              That never wrote to me,-
            The simple news that Nature told,
              With tender majesty.

"World" and "told" comprise a consonant rhyme.

                 His is the halcyon table
                      That never seats but one,
                    And whatsoever is consumed
                      The same amounts remain.

                             from "Life," III, Emily Dickinson (~1890)

"One" and "remain" comprise a consonant rhyme.

           Drowning is not so pitiful
              As the attempt to rise.
            Three times, 't is said, a sinking man
              Comes up to face the skies,
            And then declines forever
              To that abhorred abode
            Where hope and he part company,-
              For he is grasped of God.
            The Maker's cordial visage,
              However good to see,
            Is shunned, we must admit it,
              Like an adversity.

                             from "Life," IX, Emily Dickinson (~1890)

"Abode" and "God" is a striking consonant rhyme.
 

Assonance:  In assonance, when a rhyme repeats, only the vowel sound is heard.  Only poets with fabulous ears, such as Dickinson or Bogan, do this well.

          Doubtless we should deem superfluous
              Many sciences
            Not pursued by learned angels
              In scholastic skies!

                        from "Life," XI, Emily Dickinson (~1890)

"Sciences" and "skies" is a lovely assonant rhyme.
 

     Heaven is what I cannot reach!
              The apple on the tree,
            Provided it do hopeless hang,
              That "heaven" is, to me.

"Reach" and" "tree" form an assonant rhyme, "tree" and "me" a perfect rhyme.

            Life, and Death, and Giants
            Such as these, are still.
            Minor apparatus, hopper of the mill,
            Beetle at the candle,
              Or a fife's small fame,
            Maintain by accident
              That they proclaim.

                    from "Life", XXXIX, Emily Dickinson (~1890)

This contains both assonant and consonant rhymes, an assonant rhyme between "Giants" and "candle," and a consonant rhyme between "Giants" and "accident."  The first might be disputed but the line ending sounds in this poem are closely related. Dickinson is probably the master of assonant rhyme in English over the last century and a half.  It takes careful thought and a well-trained ear to do this well.

While the author hunts down more, here are some simple ones to play with:
                     aught, awe
                    note, row
                    sleep, flea
                    conflate, delay
                    deny, delight
                    soufflé, berate
                    hell hole, Jello mold

Okay, the last one's shaky.

Falling rhyme:    In this scheme, "bells ring" might rhyme with "happening," where the rhyme sound is on a (relatively) unstressed syllable.   Useful for de-emphasizing both a rhyme and a line ending.
 

 Nelson was once Britannia's god of war,
    And still should be so, but the tide is turn'd;
  There 's no more to be said of Trafalgar,
    'T is with our hero quietly inurn'd;
  Because the army 's grown more popular,
    At which the naval people are concern'd;

"of war" and "Trafalgar" are a pretty good falling rhyme, at least now where the accent falls on "fal..." in Trafalgar.  Were they in Byron's time?

You are far more likely to find falling rhyme in poets of the past seventy-five years than before.

Much of what's described as slant rhyme could easily be described as either assonant or consonant rhyme.  Further, not a little of what we perceive as slant rhyme in poets from centuries back is due to shifts of stress and sound that have occurred in the evolution of dialects and accents.  (The sound of "ie" and "y" were nearly the same for Donne, for instance.)

Distant or off rhyme is all the rage nowadays, often so distant that nothing remains of the rhyme but the author's unexpressed intention.  See Craig Raine.  But where it's effective, such "off rhymes" can usually be described by one of the four varieties mentioned above.

Rhyme begins as a child's game, and those who don't escape that game tend to write rhymes that are easy, even trite, which may be perfectly all right in a musical lyric, comic poetry or, occasionally, in serious poetry.  But someone with a highly evolved sense of rhyme, such as Cole Porter, Dr. Salemi or Richard Moore, can contrive exotic double, triple and quadruple rhymes without sounding precious or unhinged.  (See excerpts from Moore's Mouse Whole in archives.)

Rhyme should be shaped to follow a poem's tone and relative seriousness or humor carefully.  A trap of rhyme, the sense of lurching, stop and start lines, is wonderfully parodied by Shakespeare in the play-within-the-play in Hamlet.

    But woe is me, you are so sick of late,
    So far from cheer and from your former state,
    That I distrust you. Yet, though I distrust,
    Discomfort you my lord it nothing must.
    For women's fear and love holds quantity,
    In neither aught, or in extremity.
    Now what my love is, proof hath made you know,
    And as my love is sized, my fear is so.... (etc)

The fragment from the Player Queen in the play-within-the-play, said aloud, borders on idiocy.  Not only does the playwright's method separate the play-within from the play about Hamlet and the Danish court, but it vividly displays some of the worst habits we can fall into when writing metrical and rhyming verse.  The couplets clank along, each one with strictly  masculine line endings, the inversions obviously forced and silly ("discomfort you my lord it nothing must").  And, as the scene progresses, the Player King's and Player Queen's lines and couplets fall apart into epigrams which force the plot to drag from couplet to couplet (and not so incidentally increase the tension among the courtiers "watching" the play-within-the-play).  I have been told by a pretty good actor that these lines are harder to memorize and play than any others in Hamlet, including those in prose,  precisely because their singsong rigidity fragments the speeches.  In fact, they often come perilously close to making no sense.

That can be avoided by an artful alternation of the rhyme's part of speech, a verb in the first instance, a noun in the next, an adjective, and so on.  Such art will pull a listener or reader through what might otherwise be a dead stop, as well as past too much consideration of how trite a rhyme is.   Here's a small piece of Frost's "Blueberries" from North of Boston:

    "You ought to have seen what I saw on my way
       To the village, through Mortenson's pasture to-day:
       Blueberries as big as the end of your thumb,
       Real sky-blue, and heavy, and ready to drum
       In the cavernous pail of the first one to come!
       And all ripe together, not some of them green
       And some of them ripe! You ought to have seen

He doesn't change part of speech with every rhyme but does so often enough, in combination with enjambment, to keep the piece moving.  Brodsky was a master of this, though his rhymes were sometimes maddeningly trite.

Another tactic, often used by Auden, and by a myriad of others, is internal rhyme, where the first instance is repeated in the middle or beginning of the next line, further varying both expectations and results of using rhymed, metrical lines.

Among some poets, such as Keats and Dickinson, what might be called modulation of the rhymes (similar to modulation of keys) seems to be at play (we can't say for certain unless they tell us), with a mixture of exact, assonant and consonant rhymes that seem to resolve, often on a repetition of an initial set of rhymes.  Not least is that it's challenging to try; but skillful modulation of rhyme sounds can be as satisfying as the modulation of keys in a piece of music, moving from first statement to a series of variations to a recapitulation.  Keats's odes show something like this, as do dozens of poems by Dickinson.   Dante's canzones -- at least he claimed this, and it seems to be borne out by some in Vita Nuovo -- modulated rhyme schemes by using related, but not the same, rhymes in successive stanzas.  You can approach this in English, but barely, but can get something closer to this effect by using assonant and consonant rhymes based on the first stanza's rhymes in successive stanzas, with a recapitulation of the original rhymes in the last.  As with the composition of music, what you do with a given toolset can offer more most exciting originality than absolute novelty.

Rhyme begins and ends with the ear.  You must hear rhyme to use it well.  People who don't have an ear will try to "rhyme" laugh with Waugh,  what might be described as a purely visual rhyme,. instead of the former with the perfect giraffe, the consonant puff, or the assonant daft, and the latter with the perfect spa or the assonant not.  Listen; don't scan.

As a last remark, the application of rhyme in music is quite different, because duration is up to the composer of the music.  Duration can change exact rhymes into near rhymes or so overwhelm the ear that a rhyme may sound like less than an echo. But you have to work with or be a composer to play on that field.
 

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