Developer, Former MVP, now at Microsoft - Best of 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013
Some Windows Phone 7 owners already have the NoDo update, and others are getting it soon. This update brings, among other things, copy & paste support for text boxes. The user taps on a piece of text (and can drag in either direction to select more/less words), a popup icon appears that when tapped copies the text to the clipboard, and then at any app that shows the soft input panel there is an icon option to paste the copied text into the associated textbox. For more read this 'how to'. Note that there is no programmatic access to the clipboard, only the end user experience I just summarized, so there is nothing you need to do for your app's textboxes to support copy & paste: it just works.
The only issue may be if in your app you use static TextBlock controls, for which the copy support will not appear, of course. That was the case with my Translator by Moth app where the translated text appears in a TextBlock.
So, I wanted the user to be able to copy directly from the translated text (without offering an editable TextBox for an area where user input does not make sense). Take a look at a screenshot of my app before I made any changes to it. I then made changes to it preserving the look and feel, yet with additional copy support (see screenshot on the right)!
So how did I achieve that? Simply by using my co-author's template (thanks Peter!): Copyable TextBlock for Windows Phone.
(aside: in my app even without this change there is a workaround, the user could use the "swap" feature to swap the source and target, so they can copy from the text box)