1. Verify that the bug occurs in the latest available (possibly unstable/nightly) build.
2. Submit the bug to the relevant browser/engine bug tracking system (see below). If you aren't confident about how to write a good bug report, read [1] first.
3. Write a testcase. If at all possible this should be minimal (i.e. not using the whole of jQuery/Angular/whatever) and should certainly be clear (e.g. no minimised code unless it is actually needed to reproduce the issue).
4. If the bug is a conformance bug rather than a QoI issue, submit your testcase to the W3C [2] so that it can be incorporated into the standard testsuite and automatically run by all browser vendors. To do this the test needs to be written in a standard format [3] and submitted using the process at [4]. At the moment the documentation is a bit sucky, but there is a big revamp in the works [5], so that should improve in the near future.
That last one may sound like a lot of effort, but testing is the only way that we will end up with a web platform that is both technologically competitive and open both in spirit and in practice i.e. with multiple interoperable implementations. People put a huge amount of time into working around, and complaining about, cross browser issues. Devoting a small fraction of that time to submitting regression tests instead would dramatically decrease the number of problems in the future. For example submitting a test for canvas clipping both makes it likely that IE will be fixed in the near future, and also ensures that it won't then just regress again one version later.
It seems that it would be in the self-interest of big projects like jQuery to commit to creating a test for each time they have to work around some browser bug, so they can expediate fixing the issue, track which browser versions have the bug, and remove the workaround — and thus reduce their code complexity — as soon as the test passes in all the browsers they are targeting. However this is not just something that matters to big projects. If you have run across a bug that is making it difficult to implement something, you are likely the best person in the world to write a testcase for that issue.
If you are trying to submit bug reports and tests and need help, please ask on #whatwg on Freenode irc or #testing on irc.w3.org [6].
Bug trackers:
Gecko: https://bugzilla.mozilla.org/
WebKit: https://bugs.webkit.org/
Blink: http://crbug.com/
Trident: http://connect.microsoft.com/IE
Opera: https://bugs.opera.com/wizard/
[1] http://fantasai.inkedblade.net/style/talks/filing-good-bugs/[2] https://github.com/w3c/web-platform-tests
[3] https://sites.google.com/site/forthenewbies/home/writing
[4] http://testthewebforward.org/resources/github_test_submissio...