<p>How do I let Microsoft know about a problem I've found in one of their core library routines? Do they have a central repository to report these things?</p>

<p>I am not a member of Microsoft Development Network (MSDN).</p>

<p>Or should I even bother?</p>

There is no official way to report bugs to Microsoft for an end-user. If you are participating in a beta program for an upcoming release, the beta program includes a bug-reporting channel. Otherwise, if the bug causes problems that you want to get resolved, you can call Microsoft support, and they will help you solving the problem (be it by providing a patch, or a work-around); if the problem turns out to be caused by a bug indeed, they will refund the costs of the support call.
Microsoft does have a central repository (perhaps separate ones per product), but this repository is not accessible for the general public.