Attempted proofs of Fermat's Last theorem by cranks used to be so common that Edmund Landau, a German mathematician, had a form letter for them: “Dear Sir/Madam: Your proof of Fermat’s Last Theorem has been received. The first mistake is on page _____, line _____.”
I have sometimes seen notes on the pages of prominent mathematicians that they don't have time to examine unsolicited attempts at major problems by amateurs so there must still be a good number. I also have known someone whose sibling is a software engineer and whose hobby is trying to resolve P ? NP.
I heard a talk by one of the main vetters of the P?NP problem (which itself was underwhelming; clearly presentations were not his strong suit) and he said there were vast numbers of attempted proofs, only a small fraction of which are even treated seriously, and none of which are considered for long enough to be considered "close" to complete. He seemed to believe that the only significant hope for the problem was in the hands of Ketan Mulmuley at UChicago (http://www.cs.uchicago.edu/people/mulmuley), but realistically a solution would not be found in our lifetimes.
This is not only interesting for that specific problem, but it was revealing in how many amateurs (including those with doctorates) who are completely over their head at the edge of our knowledge—the people who understand it well enough to evaluate an approach adequately don't submit because they understand the difficulty in ways most don't.