Rhyming lines? Nope. Regular meter? Not that we can see. While most of the lines do look like they are around the same length (7-8 syllables), we also get these long, straggly lines randomly throughout the poem (for example, lines 7, 11, and 13). We're not sure that any clear, logical reason exists for why these lines are so long, or why they appear where they do.
But before we just label "Silence" as free verse, we should notice that the poem has 14 lines. 14 lines? That immediately makes us think of a sonnet. Sure, unlike "Silence," traditional sonnets have a rhyme scheme and follow a metrical pattern (usually iambic pentameter; think Shakespeare or John Donne), but they are also almost always 14 lines long. Also, many sonnets have a "turn" that occurs in the last two lines – a sort of surprise ending, much like the punch line to a joke.
Like a sonnet, "Silence" also has a turn at the end. Almost the entire poem is a single quotation from the speaker's father, but in the second-to-last line (line 13), the speaker interrupts: "Nor was he insincere in saying…," setting us up for the punch line at the end: "Inns are not residences" (line 14). So while Moore's poem doesn't exactly fit our idea of what most sonnets are like, it borrows a couple key characteristics.