I have updated over 50 sites without error. From the information you have provided it is not possible to help you

Application Instantiation Error

Perhaps the error is coming from a 3rd party extension?

Perhaps you could give a little more information about your error?

Just once I'd like Joomla to issue an update that actually wo

Perhaps you could help with testing Beta and RC versions before the update is released rather than posting stupid comments like that

I too have updated sites to 3.8 without any problems

I try update j3.7.5 to 3.8
In adminpanel i have the critical error:
Call to undefined method JAdminCssMenu::addChild()
i use a 3rd party extension - Zoo
in front of i have problem too

can you test without 3rd extensions?

OK
I disabled the extension
and again tried to update Joomla
update is successful but now I get this notice:

Notice: Constant JROUTER_MODE_RAW already defined in /home/alex/www/delfa.loc/libraries/cms/router/router.php on line 17
Notice: Constant JROUTER_MODE_SEF already defined in /home/alex/www/delfa.loc/libraries/cms/router/router.php on line 24

in adminpanel I can not get
I keep getting redirected

If this is on 3.8 then your upgrade was not successful as the libraries/cms/router/router.php file should have been deleted.