Apparently, this seems to be caused by a critical patch to IE which was applied over the weekend - http://technet.microsoft.com/en-us/security/bulletin/ms13-028. In order to solve this problem, the current workaround is to proceed with the following:

Workaround 1: Reset the IE settings

1. Click on Tools/Internet Options and go to the advanced tab (At the bottom of that window there will be a Reset button).
2. Click this button to bring up the reset window.
3. Then click the reset button on the bottom of that window.
4. Close IE and open it again. JIRA should be working fine.

Workaround 2:
Use other browsers or use different computers.