To troubleshoot, I had created custom module titled demo which isnt appearing too when assigned for specific pages although it appears when assigned for all pages so the issue persists for other modules too. As the specific issue relates to core joomla functionality or module positions which are told by specific template to the Joomla! application where to place output from modules assigned, You may kindly post for the same in Joomla forums or RT forums whose template has been assigned as default for the site
You may also replicate the similar issue with aforesaid module
Also, To append, You may try the following :
1) Kindly try advanced module manager to see if it helps
2) Assigning default joomla template and check if the core joomla functionality for module is working fine by assigning specific module for available positions in the default template