The first workaround is to clear browser cache and Refresh the page. Actually, "Refresh" and "Reload" the page - are 2 different actions! You can reload the page using F5, refresh - using Control+F5. In last case the browser's cache cleans and the full page loads successfully. So it seems that the problem is related to incorrect caching of the web page objects. It's the nature of the problem. Incorrect loading/functioning of JavaScript is the aftermath. After the problem was solved I added settings to web server configuration that change the Cache-Control header of the web page so that it prevents the browser from caching it. I added the following parameters to Cache-Control:
no-store, no-cache, must-revalidate, post-check=0, pre-check=0

Such modification works on all browsers I tested and now my Moodle works fine and stable without menu problems! But of course such solution is temporary and completely turns off cache. So it results in degrade of Moodle's loading speed, efficiency and performance.

Tested in Moodle 2.2.1+, build 07.02.2012. Problem detected in the following Windows web browsers: Google Chrome ver. 17.0.963.56 m, Firefox 10.0.2, Opera/9.80 (Windows NT 6.0; U; en) Presto/2.10.229 Version/11.61, Explorer 9.0.8112.16421, Safari 5.1.2 (7534.52.7). Usually it's possible to reproduce the problem by manually reloading the web page (clicking "Reload" button in web browser). When clicking the grey menu items, e.g. "Administration" menu item - nothing happens. The menu item is supposed to expand. Complete list of tested MOODLE versions:
1) Moodle 2.2.1+, build 07.02.2012
2) Moodle 2.3.1+ stable, version 2012062501.11, build 20120823.
3) Moodle 2.4.1+ stable, build 20130418.