Access to math papers I graduated with my PhD in math two years ago, and I am no longer affiliated with a university. However I am still actively researching. Often it's possible to get papers on arxiv or directly from the author's website, but sometimes it isn't (particularly for older papers). Public libraries, at least around here, don't have subscriptions to mathscinet. Do I really have to pay if I want these papers?
Mathematicians will often send you their preprints (not the work in progress), if not already on their pages, if you ask kindly. I suggest to title the email with "Paper request: [Paper title], [Journal name], [Year]" for an efficient query.

Back in 1996, I was working my Msc on a subtopic of number theory (diophantine approximations and continued fractions of polynomial roots in finite fields). I was away from my university, and working without a university student card. I was spending a lot of time at Institut Henri Poincaré, photocopying articles from mathematical journals for my research. I badly needed a conference paper: A. J. van der Poorten, Continued fractions of formal power series, Advances in number theory, Proc. of the 3rd conf. of the CNTA, 1991. No university around had these proceedings, and Australia was too far.

"Alf" was a big name in the field (to me at least). And email and WWW were not very widespread in France. Yet I found his email, and spent about 1 hour to write the most polite email I could, in the  afternoon. The next morning, the postscript was in my inbox, with a nice answer by A. J.: "thank you for your interest in my work, good luck with your study."

This gesture both:


permitted me to work on myy thesis,
boosted my confidence (as a low-level student) in approaching persons.


which helps me a lot now. I am still thankful to him now, and often think about  him, even if I never met him. So far, mathematicians have always answered my requests, sometimes even with a scan and standard mail (with nice  stamps) when the  paper does not exist in electronic form.

Now, a lot of papers are available online. I insist one should browse researcher's webpage pages (instead of relying only on web search): sometimes, old papers are present in formats (dvi, ps.gz) that are not always indexed.

Finally, this could lead to discussions and, who knows, collaborations. 

If everything else fails, different  options exist. One with  twitter hashtags has made the  news, I only mention it for the sake of information, and do not endorse the legal aspects.