Robert Browning uses a number of interesting figurative and sound devices in "My Last Duchess." The sound devices include alliteration, assonance, and consonance; here are some examples of each:

Alliteration: "Perhaps Fra Pandolf," "such stuff," "cause enough for calling up," "she liked whate'er she looked on, and her looks went everywhere," "dropping of daylight," "skill in speech," "disgusts me; here you miss, or there exceed the mark," "some stooping," and "notice Neptune." 

Consonance: "Pandolf by design, for never read strangers like you that pictured countenance," "curtain I have drawn," "ask me if they durst," "hope to reproduce," "half-flush," "liked whate'er she looked on," "nine-hundred years-old name," "herself be lessoned so, nor plainly set her wits," "known munificence is ample warrant that no just pretense of mine," and "Innsbruck cast in bronze." 

Assonance: "Pandolf's hands," "never read," "not the first are you to turn," "mantle laps," and "forsooth, and made excuse--e'en then would be some stooping, and I choose never to stoop." 

Here are some examples of figurative language Browning uses in the poem:

Personification: "...there she stands. Will 't please you sit and look at her?" Rather than referring to the painting as "it," the Duke speaks as if the painting is the actual Duchess.

Pun: "I said Fra Pandolf by design." The Duke uses a play on words when discussing the artist, since "design" is an artistic term, but he means "on purpose." 

Imagery: "Faint half-flush that dies along her throat" and "bough of cherries" both bring images to the reader's eyes.

Syncechdoche: "She had a heart ... too soon made glad." A part of the Duchess is used to represent her self or her emotional being.

Metonymy: "My gift of a nine-hundred-years old name." The Duke uses something associated with his social position, his name, to represent his social position.

Understatement: "Even had you skill in speech--(which I have not)." The Duke is obviously a skilled speaker--this is ironic. 

Metaphor: "Here you miss, or there exceed the mark." This could also be an idiom, but it calls to mind archery. "E'en then would be some stooping." The Duke compares instructing the Duchess to stooping, a physical posture that puts him in a lower position.

Ambiguity: "I gave commands; then all smiles stopped together." Also an understatement, these words imply the Duchess was executed but leave the interpretation open.

Elevated Diction: "The Count your master's known munificence is ample warrant that no just pretense of mine for dowry will be disallowed." This language seeks to impress by using fancy words and syntax. It also includes litotes, an understatement that uses a double negative: "no ... disallowed."

Symbol: "Notice Neptune, though, taming a seahorse." The sculpture symbolizes the Duke's dominance over weaker people, especially the Duchess.