Do you think the poet's sigh indicates regret or relief in "The Road not Taken?"

This could be argued either way, but I am going to say the poet's sigh indicates regret, and I will explain my reasons for saying so.

The most common interpretation of the last line of the poem is that the poet is probably content to have taken the road less travelled. That is why we often advise people—particularly young people—to take the "road less travelled," or the more unconventional choice, to live a rich life. The poem itself, however, is ambiguous on this point, meaning one can read the last lines as either approving or disapproving of the less-travelled road:

I took the one less traveled by,

And that has made all the difference.

Was the difference positive or negative? We don't know. We only know it has made "all" the difference, for better or for worse.

Assuming taking this road made a negative difference and the poet knows he definitely should have taken the other road, then the "sigh" would surely be one of regret. The poet says at the beginning of the last stanza,

I shall be telling this with a sigh

somewhere ages and ages hence

But what if the story had a happy ending, and the road less travelled made a positive difference? Wouldn't the sigh then be one of relief?

Not necessarily, in the context of the poem. The poet is imagining telling this story with "a sigh" at some time "ages and ages hence." In other words, he imagines sighing in the far future—not next year, but "ages and ages" from now. He will be in middle or old age and looking back over his life when he sighs. We know from psychology that one of the attributes of middle- and old-age is regret. No matter how good our lives have been and how many good decisions we have made, there is always something to regret. That is simply part of being a human being; we can't do it all, and one path pursued, no matter how wonderful, means another path is not pursued. The poet recognizes this truth when he says earlier in the poem that, much as he would like to return and take the other path to see where it would lead, "I doubted if I should ever come back." It is directly following that line that he says, "I shall be telling this with a sigh." The sigh suggests the narrator realizes that as he ages he will regret roads not taken, paths not pursued. This might be a bittersweet regret, but it will be regret all the same. This regret underlies so much poetry, a lament that we are not immortal; our time runs out and our choices form us in one way rather than another.