I posted this answer because I didn't find an answer which offered all possible solutions. Consider this answer as a pleasant checklist which should help you to fix this issue.
a) Permalinks update. The following steps rebuild the internal settings.
a) Login to your WordPress dashboard.
b) Go to Settings -> Permalinks.
c) Don't make any changes, just click "Save Changes" button.
b) .htaccess
needs the settings for the rewrite engine:
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
c) Also, due to an error in the Apache log file the following entry was needed as well in the your-wordpress.conf
Apache file which is mostly in the /etc/apache2/vhostd.d
directory.
<Directory your-wordpress-directory>
Options Indexes FollowSymLinks
AllowOverride All
Order allow,deny
Allow from all
</Directory>
d) Assure mod_rewrite
is enabled
a2enmod rewrite
Restart Apache