How does "The Second Coming" by William Butler Yeats exemplify the themes and ideas of Modernism?

"The Second Coming" is an example of a modernist poem because of its themes of despair and foreboding.

Yeats wrote this poem in 1919, a period in the direct aftermath of World War I and the Russian Revolution, when the old norms of European life seemed to have collapsed. World War I was especially shocking to those who experienced it, because it challenged the idea, prevalent in the nineteenth century, that Europe was a civilized continent that had ushered in an era of peace and prosperity. Instead, Europeans were faced with a barbaric, costly bloodbath that many felt was pointless and that left large numbers of people stunned and deeply traumatized.

This sense of loss and despair from trying to understand what went wrong is integral to modernist poetry and a sharp contrast to the optimism of Victorian poets like Tennyson. The sense that a frightening new era has descended permeates "The Second Coming." Rather than envisioning a bright future, the poem's narrator uses fearful imagery to describe what is to come, described in lines such as:

Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed

What are the modernist and post-modernist elements in 'The Second Coming'?

If we agree that modernism in poetry can be characterized generally by the poet's alienation from his society, the avoidance of naturalistic representation and the use of fragmentary images, often in odd juxtaposition, then I think we can argue that Yeats's "The Second Coming" is an example of modernism.  Whether the poem exhibits "post-modernism" is not as clear.

Clearly, Yeats is describing a world that has gone completely out of control.  Even something as natural as the control of a falcon by the falconer has gone awry: the falcon, rather than spiraling down either to its prey or to the falconer's arm, is spiraling outward and is no longer under any control because he cannot even hear the falconer's whistles.

Then, we have a series of images in odd conjunction, which is characteristic of modernism: a bloody tide spreading itself; the drowning of innocence; and the greatest image of a world turned upside down--good people who have lost their ability to believe in anything, and bad people who believe passionately in very bad things.

One aspect generally thought to illustrate modernism in poetry is the poet's "cultural dislocation," the sense that normalcy is no longer the norm, that society is beginning to implode.  Yeats's second stanza expresses this aspect fully.  He imagines a monstrous being from the desert, "a shape with lion body and the head of a man" (clearly, the Sphinx), is slowly making its way to one of the most culturally significant places on earth: the beast "slouches towards Bethlehem to be born."  This is the ultimate expression of the poet's despair, an unmistakable aspect of modernism.

If we take post-modernism to be essentially a continuation of modernism's alienation from society and view of life as an unconnected series of fragments, then "A Second Coming" exhibits those elements.  Keep in mind, however, that critics are still trying to decide exactly what post-modernism is, and I don't think it's useful to impute post-modernism to this poem.  It is modern enough.