Special:ListFiles doesn't find anything if the search term contains german umlauts (the letters äöüÄÖÜß) - for example, searching for "Möhre" will give zero results, but I have files like "Möhre-Wilde-Blüte.jpg" and "Pfalz-Möhren-1.jpg".

This is a duplicate of T34207 where no patch was written, but a valid workaround exists (at least for REL1_31) and hopefully could be solved in the next REL1_35 release. Here the workaround (source https://www.mediawiki.org/wiki/Topic:Pwjch65m0lbr7n8e):

Edit the file includes/specials/pagers/ImageListPager.php and replace the line:

$conds[] = 'LOWER(' . $prefix . '_name)' .

with

$conds[] = 'CONVERT(' . $prefix . '_name USING utf8)' .