Interesting question. I hadn't though to look, so I did.
There don't appear to be any apps specifically designed to help with writing poetry. Any developers out there with a literate bent?
There are a couple apps that promise to help you share your poetry with like enthusiasts. As a social medium it is harmless pastime, but it's not a very good environment for improving your own writing.
There are loads of apps that promise you a cornucopia of poetry in various styles. I'm kind of cynical, so I'd have to be convinced of their quality before reading them, even if they are free. Junk is easy to find.
The best app for writing poetry is going to be a simple writing app. I'd probably pick a note taking app with email export or cloud syncing. You could also choose a fuller featured app like QuickOffice. Really, anything will do. You can adjust the layout on the computer later if needed.
For reading I'd try regular book reading apps: Kindle, Boarders, Nook. There are several apps that offer large libraries of free, small independent publishers, and self published books. Lots of poetry available in the stores, both paid and free.
Sorry. Best I can do. I'm going to keep an eye on your thread just incase someone has a better answer. I wouldn't mind a poetry friendly writing app myself. Just because I don't care to read junk poetry doesn't mean I'm immune to making my own.