How do I clear cache using an SQL query? After a module update, my site has become unusable, and only shows a php error message.
I tried circumventing the problem with drush cc, but that didn't help.

Also tried with a custom php script but that has issues finding my DRUPAL_ROOT

I just want to clear the cache tables from Drupal 7 directly on the mysql server, but I'm unsure which tables should be cleared for that and especially which I should not clear.

Do I just need to clear all the [SITE-PREFIX_]cache* tables?
Yes, you can simply clear (TRUNCATE) all cache* tables. 

This worked well for me:

TRUNCATE TABLE cache;
TRUNCATE TABLE cache_block;
TRUNCATE TABLE cache_bootstrap;
TRUNCATE TABLE cache_field;
TRUNCATE TABLE cache_filter;
TRUNCATE TABLE cache_form;
TRUNCATE TABLE cache_image;
TRUNCATE TABLE cache_menu;
TRUNCATE TABLE cache_page;
TRUNCATE TABLE cache_path;
TRUNCATE TABLE cache_token;
TRUNCATE TABLE cache_update;


If you use drush, run drush sql-cli and paste the above in there.

These commands might not clear all cache tables of your specific site but it can help fix some errors. Afterwards you can try drush cc all to clear the rest.