after Update to Joomla 3.7.0: 1146 Table 'dbXXXXX.#__' doesn't

I can't reproduce your issue neither with php 5.6 nor with 7.1.
A little question did you change your db to PDO? Such change will produce problems!

Hi Georg,

I don't have a concrete idea because this effect is very strange but I think it has nothings to do with JEM but more with Joomla or upgrading it to 3.7.0.

Within JEM we never (except Import classes) have a string "#__" without directly followed table name. Also this special token has to be replaces by site's table prefix by Joomla's DB classes. So the DB will never see "#__".
So I think there is something wrong with Joomla files.

If you have any more information, ideally a (php) filename, or the full SQL command causing the error - maybe somewhere in a log file - it could be helpful to limit the pieces of code involved.
Do you talk about backend or frontend? Is this effect on JEM only? Or have you also problems on articles, contacts or other stuff? Do you have additional plugins?
A complete backup (e.g. as Akeeba Backup jpa file) would be useful - but I really don't want to get any private data of anyone. So if it's a live site better you don't provide such a file.

Another idea could be to copy all the files (not db tables) to your local pc and compare them with Joomla install package (especially the libraries/ part).