In order to help you we need to know your specific computing environment details. Please work through what follows and report back.
1. If you are not running one of the browsers and versions listed at http://browsehappy.com/ please try upgrading or switching first.
2. If you are using an IE browser in compatibility mode please disable that mode. http://www.sevenforums.com/tutorials/1196-internet-explorer-compatibility-view-turn-off.html
3. If you are running one of the browsers and versions listed at http://browsehappy.com/
(a) Try clearing your browser’s cache http://www.google.com/support/accounts/bin/answer.py?hl=en&answer=32050
If applicable see here for instructions on how to clear the cache for Chrome and Safari on the iPad:
http://browsers.about.com/od/howtousemobilebrowser1/ss/safari-ipad-clear-cache.htm
http://browsers.about.com/od/howtousemobilebrowser1/ss/How-To-Clear-Browsing-Data-In-Chrome-For-Ipad_6.htm
(b) Make sure that you have both third-party cookies enabled and JavaScript enabled
http://en.support.wordpress.com/third-party-cookies/
http://www.google.com/adsense/support/bin/answer.py?answer=12654
(c) Disable all browser extensions or add-ons temporarily.
(d) Try enabling HTTPS: http://en.support.wordpress.com/https/
Add both *.wordpress.com and *.wp.com to your list of trusted sites under the Security tab. Make sure to uncheck the box for “Require server verification (https:) for all sites in this zone.” http://en.support.wordpress.com/https/#frequently-asked-questions
4. Try using another browser.
5. If you are not successful let us know:
(a) Exactly what kind of device you are using to connect to the internet and to WordPress.com.
(b) Exactly which browser (and version of it) you’re using by checking here if necessary http://www.whatbrowser.org/
(c) And what any error reports you receive state.
All threads with the same issue are tagged for Staff. What's best is to simply subscribe to the thread with the most comments in it here http://en.forums.wordpress.com/topic/disappearing-links?replies=42 and wait for a Staff response.