The renewal of this 2010 question asks for more recent NSF math grant proposals.

A collection of proposals from 2000 through **2015** is at <A HREF="http://www.math.stonybrook.edu/~bishop/vita/">Christopher Bishop's</A> home page.

This <A HREF="http://people.math.gatech.edu/~dmargalit7/tsr/grants.shtml">site</A> maintained by Dan Margalit has one example of a successful <A HREF="http://people.math.gatech.edu/~dmargalit7/tsr/career.pdf">NSF math proposal</A>, and in addition contains resources on how to write such a proposal (notably the sections on intellectual merit and broader impact).

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The renewed OP also notes broken links: some of these are on the Wayback Archive:

 - <A HREF="http://web.archive.org/web/20120511024054/http://modular.math.washington.edu/grants/screms/">SCREMS:
   The Computational Frontiers of Number Theory, Representation Theory,
   and Mathematical Physics</A>
   
 - <A HREF="http://web.archive.org/web/20140920210817/http://faculty.uml.edu/jpropp/Quasirandom.pdf">James
   Propp (1)</A> and <A
   HREF="http://faculty.uml.edu/jpropp/Entire.pdf">(2)</A>