Prosody  is the most important characteristic of poetry; rhythm distinguishes poetry from prose fiction. Metrical patterns give physical and emotional meaning. Remember, though, that not all text rendered in verse form is poetry - just because it rhymes and jingles doesn't constitute poetry. Prosody is a technical term used in linguistics and poetry to describe the patterns, rhythms or meters of a language. Prosody can refer to the rules for the pronunciation of a language as well as its versification. Poet chooses appropriate meters and verse forms to help distinguish one character, racial group, idea, or thing from another and to convey information about their stature and stance. Given the ways in which poet uses versification to signify tone, context, and aspects of character and society, readers of his poetry cannot hope to approach a full understanding of his work without paying due attention to one of the primary ways in which poet  encoded his poetic meanings

I have been hired to recite verses from the famous Shelley poem much quoted by Jeremy Corbyn.   I am not used to speaking in public, and am certainly not going to declaim any of those verses in the poem that do not scan!   I greatly enjoyed Latin verse at school, no doubt because it scanned and had easy rhythms suitable for public recitation by uneducated persons.  

Prosody is the musical side of language. Hence its importance to literature, both in prose and in poetry, and of course in songs. The phenomenon of the VERSE, as opposed to the SENTENCE, is constitutive to poetry. But prosody is also a grounding aspect of (prose) rhetoric. The human mind is rhythmic; both thoughts and emotions are pulsating bodily and mental emanations, and the intertwining of sentence rhythm and metric rhythm in poetry and song yields something like  'emotional thinking'.

I agree with the previous comments. Prosody is important because poetry sparks elements of language that contribute to stimulate the emotional zones.

Poetry is largely prosody, therefore, the question is not 'What is the importance of Prosody in the study of poetry?' But how can the study of poetry contribute to a better understanding of prosody?

In certain theories of linguistics particularly in the semantic field, prosody is part of building the over all picture of feelings/emotions of the writer. Very often this form of prosody is referred to as waves which manifest themselves in the text. 

 Tra utopia e realtà  è così che io definisco la descrizione del testo poetico infatti interagiscono nella descrizione del verso la linguistica la neurolinguistica ecc. Ascoltando un melodioso verso di Dante Alighiero "il maestro" del verso non posso che essere  riverente  alla sua ottemperanza ma.... Egli come tutti ha percepito nel suo profondo delle emozioni che gli hanno fatto generare versi e ritmi "frutto delle sue competenze " Io oggi da linguistica non posso non scansionare le sue morfie allitterative e riprodurle con strumenti elettronici , ma guarda casa la prosodia e la metrica c'è perchè il parlante ha innato la ricevenza e la descrizione dei linguaggi  

Prosody is only valuable if it's intelligent. Review scripts from hit songs by Stevens, Elton, or even Lennon... their common denominator, even when rhyme is absent, is wisdom in the words.

I am currently working on an analysis of Leonard Cohen's Hallelujah song. It holds together love and religion by the chords that the first verse mentions. The prosodic pattern of the text and the meter, melody, and harmonics of the music are closely related. – Maybe prosody is in general a sort of 'potential music' in the syntactic, lexical, and rhetorical patterns of the text. In that case, we need to study the relations between musical structures and human bodily emotional expressions in order to better understand prosody. (As Xosé says above.)

You may be interested in Jeremy Hardy's interpretation of Hallelujah as performed by George Formby (on This week's I'm Sorry I haven't a Clue). Listen to http://www.bbc.co.uk/programmes/b08yncyj from 24:10 minutes in. Definitely different prosodics.

Good suggestion sir

Uh, not available, says site.   :(

Love made me poet,

And this I writ;

My heart did do it,

And not my wit.

Lady Tanfield

In The London Review of Book, 5 August 2010, a reader, S. J. Silverman, brings forth this anonymous perfect monometric epistolar sonnet, which I will reproduce without comments:

An Aeronaut to His Love

I through blue sky

fly to you. Why?

Sweet Love, feet

move so slow.