Are there examples of (successful) NSF mathematics proposals available anywhere online? As anyone who has ever applied to the NSF for a grant knows, such a proposal is a slightly odd piece of writing, not quite like anything else mathematicians are called upon to write.  As such, it's a hard thing to learn to do well; of course, the basic requirements are set out in the grant proposal guide, and explained a bit more clearly in some other sources.  But there's no substitute for reading an actual specimen.  
Now, generally this is achieved by asking some older colleague to see an old proposal, an approach that works just fine in most cases.  On the other hand, if there are any publicly available old NSF proposals online, I think finding them would be a boon to all of us preparing proposals, if only to have more data points.  Does any one know of any (I didn't have much luck on Google).
I'll just note: I'm aware that the NSF has abstracts of all funded proposals on their website, which is useful, but doesn't give much insight into, say, what people write about broader impacts.  
 A: By law NSF must make the abstracts of all funded proposals public.
They have a search tool for exploring awards: link.  You can look for every proposal that
includes, say, knot in its title (there are 29 active awards, not all in mathematics!).  Etc.
[Sorry to repeat what you knew, as indicated in your final note.]
A: The renewal of this 2010 question asks for more recent NSF math grant proposals.
A collection of proposals from 2000 through 2015 is at Christopher Bishop's home page.
This site maintained by Dan Margalit has one example of a successful NSF math proposal, and in addition contains resources on how to write such a proposal (notably the sections on intellectual merit and broader impact).

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


*

*SCREMS:
The Computational Frontiers of Number Theory, Representation Theory,
and Mathematical Physics

*James
Propp (1) and (2)
A: See Doron Zeilberger's page where he provides the proposals for several of his (approved) grants and the reviews (as noticed by Ben in his comment below) for two of them:
http://www.math.rutgers.edu/~zeilberg/grants.html
For other examples, see 
SCREMS: The Computational Frontiers of Number Theory, Representation Theory, and Mathematical Physics
(it was awarded: see here)
and the awarded grant proposals by James Propp here and here.
By the way, this presentation also could be of some use:
Writing an NSF Proposal: a PI's and a panelist's perspective
as well as the Notices article
NSF Proposal Preparation: The View of an Ex-Program Officer
A: William Stein places all of his grant proposals online here : https://wstein.org/grants/
A: Off the top of my head: http://www.math.tamu.edu/~sottile/grants/0701050.pdf
I think many recipients tend to put their proposals on their websites. 
A: There are two NSF proposals by Jesse Johnson on his website:
http://www.jessejohnson.me/nsfapp09.pdf
http://www.jessejohnson.me/nsfapp12.pdf
