Shameless selfpromotion here:

How about my font Eastburgh? ;-)

I don't know of any typefaces designed specifically for poetry, other than my Panoptica, for monospaced constraints-based verse.
Economics would seem to mitigate against it. However, Panoptica has paid for itself one way or another, over time.

I do enjoy reading it, Nick.

What other poetry-designed typefaces are there?
After all, in a world of pickle forks and caviar spoons…

Acuta features a long poem in the PDF.
http://new.myfonts.com/fonts/anatoletype/acuta-medium/

For reasons random I keep thinking of George Douros's Anaktoria when someone mentions typesetting poetry, but that one is not modern, let alone contemporary.

How'bout Hideko or Fontin?

I always used Spectrum but that was metal. Others used Lutetia, Bembo, fonts by Jan van Krimpen such as Romulus, and Romanée. These looked great in metal and if I am not mistaken cast by Enschedé. Here is the original refined drawing by van Krimpen of Romanée:

P.S. We are now way past modern and, in a certain sense, beyond postmodern as well; "contemporary" might be a more appropriate adjective…

Wow thanks everyone, this was more than expected. I can't get over the beauty of Acuta's 'g', but I do like the quirkiness of Jean's Eastburgh. It might just be what the client needs to promote his work effectively. Same goes for Panoptica , though the verses I'm setting are rather long, which may not work so well.

Certainly something to dig my teeth into...

What about http://typographica.org/2007/typeface-reviews/amalia/ ?