Because of poetry's focus on structure, formalist analysis is a good choice for poetry. Formalism looks at a work of art itself, outside of the context of culture or the biography of the author. Formalist analysis is interested in the organization of a poem and in finding paradoxes and complexity within the poem. The choice of line breaks, length, rhyme scheme and stanza type are all analyzed for meaning and purpose in formalist criticism. On the language level, a formalist analysis might focus on the author's word choices, allusions and use of figurative language. Formalist analysis pays close attention to all elements of the poem and how they fit together to create the poem as a whole.
Psychological analysis involves using Freudian or Jungian terminology and ideas to understand a poem. Terms such as archetypes, unconscious, Oedipal complex and collective unconscious are used in psychological analysis. An example of this kind of criticism would be in looking for common mythological themes (such as the "hero" or the "quest") in a poem or in looking for symbols and images that indicate unconscious desires (such as finding imagery of caves or swords, which often indicate female and male sexuality in Freudian readings). This kind of poetry analysis requires a clear understanding of the terminology involved in Freudian and Jungian analysis.
Deconstructive theory looks for clues indicating that a text means ultimately the opposite of what it claims to mean in the most obvious interpretation. For example, a poem that seems to clearly be about the love between a mother and a daughter might, under a deconstructive lens, appear to actually be about deep dislike between these two family members. Deconstructive readings look for details that contradict the central point and word choices that reveal hidden feelings. A deconstructive analysis is essentially reading for contradiction, so it is important to have the basic methods of formalist analysis before you can do a successful deconstructive reading.
Literary tradition analysis assumes that a poem is part of a long tradition of previous texts and should be analyzed against that history. When using literary tradition analysis, think about what genre or type of literature the piece falls into. If the poem is a sonnet, for example, think about the history of sonnets: How have masters of the form used the sonnet, and how does this poem fit into previous usages or vary? In addition to structural literary traditions, think about rhetorical traditions. For example. elegies are a traditional form of poem about a death. If your poem is primarily about an individual after their death, think about how your poem, even if it is not clearly labeled as an "elegy" might fit into that tradition. For this style of poetry analysis, you must have a good handle on basic literary terms and some sense of literary history.