We've got a speaker reflecting on memories and current experiences in a personal, often philosophical way, which means that for much of "The Waste Land," we're reading a dramatic monologue. What makes "The Waste Land" different from a normal dramatic monologue (like Eliot's earlier poem, "The Love Song of J. Alfred Prufrock") is that the speaker is constantly shifting between different personalities, cultures, and historical moments. This gives Eliot's poem a panoramic quality while also making it very fragmented. It's hard to keep track of who's saying what, but there's no doubt that for much of the poem, they're talking to us.
Every now and then, you'll find a rhyme or a consistent meter; but these moments are always fleeting. It's fitting, though. What good would perfect rhyme and meter do in a poem about the chaos and decay of the modern world? We get the sense that maybe the speakers trying to put together the pieces of a big, cultural puzzle, but we never quite see the overall picture that the pieces are supposed to create. And hey, maybe that picture doesn't exist anymore.
The second part of the poem starts off with a healthy and refreshing dash of blank verse (a classic English meter): "The chair she sat in, like a burnished throne, / […] / Held up by standards wrought with fruited vines" (77-79). These lines convey a proper sense of the classic beauty they're describing. But this beauty and meter quickly fall apart, eventually leading to the conversation between the two women in the pub, which seems to be just too "low class" to fit any poetic form (139-172).
In other parts of the poem, Eliot inserts popular songs from his time, but usually as examples of how low culture has overtaken the glorious rhythms of classic meters. The overall effect seems to be a poem that is constantly trying to regain a structured, refined style, but keeps getting sucked back into low culture. Kind of like how you try really hard to watch an episode of News Hour with Jim Lehrer, but you always wind up watching Nancy Grace instead.
In addition to his form and meter, Eliot pulls out almost every poetic technique in the book in order to convey his ideas about modernity in this poem. Overall, he wants to give us a sense of what it feels like to live in the 20th century, and he believes that the main feeling of this time is a sense of meaninglessness and despair, combined with a lack of closure.
How does he create that sense? By using a little thing called enjambment. That keeps every line feeling like it's unfinished. Remember the first two lines? "April is the cruellest month, breeding / Lilacs out of the dead land, stirring" (1-2) leaves us hanging on each line, with those participles dragging out the sentence. Eliot also loves to use the ol' ellipsis to convey this same feeling, as is the case in line 182, where the speaker "sat down and wept…."
But he doesn't stop there. Another brutal element of modern existence is the terrible sameness that seems to determine every day of people lives. You know, that awful feeling that life isn't going anywhere in particular? Eliot conveys this most in his description of woman chatting in the pub in lines 139-172. And he especially conveys it through his use of the refrain of "HURRY UP PLEASE ITS TIME." Every time you read this, you are reminded of the fact that this chatty Cathy has probably heard this phrase thousands of times, which means that she's probably spent a big chunk of her life wasting away the hours at the bottom of a bottle of sherry, rather than working toward any sort of goal.
With Eliot, you also get constant reminders that beauty that might still exist in the world, but unfortunately, these beautiful refrains fall on deaf ears. Remember that onomatopoeic refrain of "Jug jug jug"? That tells us that Philomela (who represents classic beauty) cannot be understood by modern people, because modern people lack the education or the good sense to recognize what they're hearing.
At its heart, this poem is a form unto itself. It's fragments, stories, allusions, and images. All these things get tossed into a poetic melting pot to invent a new form—one that Eliot finds suitable for the mess that is the modern world. The reason Eliot draws on all these poetic forms, traditions, and devices is that his poem is designed to be like a Wikipedia of sorts for Western culture (though he would've hated Wikipedia). He's hoping the diversity within the poem might help reinvigorate the lost respect for high culture that pervades modernity. That's a tall order, though, and it's your call whether or not Eliot makes this happen.