I am running BOINC 7.8.3 (x64) on Windows and recently it stopped working. Upon launch "Retrieving current status" message gets replaced by "Unable to connect to the core client", and that's that.

According to the process list the Manager does not launch Client at all. But even if I start the Client manually beforehand, I am still getting the same error message from the Manager.

So fat I have tried
1) reinstalling BOINC multiple times (uninstall files, clean Program Data, remove all registry entries, restart computer, etc),
2) checking if something is using the same port (TCP 31416), and nothing does,
3) checking the firewall (I have Kaspersky, both Client and Manager are Trusted, adding an explicit rule for port 31416 does nothing).

Any help would be appreciated.

Check stdoutdae.txt and stderrdae.txt in BOINC's data for any new messages when you start the client and post them here. The data directory should be in C:\ProgramData\BOINC if you are using default settings.

Unfortunately, I removed the entire Data folder when trying to reinstall BOINC. Since then no stdoutdae.txt or stderrdae.txt files have been created.

I was going to ask: you say you tried starting the client manually. Did it keep running, or did it stop again?

We had some problems with the client crashing in early testing under Windows 10, and eventually found some crash dump files in the (hidden) folder

C:\users\[your account name]\Appdata\Roaming\BOINC

You probably haven't wiped that, so it might be worth a look.

Before the complete reinstall Client was running just fine, but the Manager still could not connect to it. Of course, now when the Data folder is gone the Client has nothing to do, but the Client process does not crash.

The Roaming folder has been cleaned out as well, I am afraid. It contains a few files now, but all of them are blank.
I was absolutely sure that if I do a really "clean" install the problem would go away. In retrospect it was not the wisest decision.

BOINC has never crashed on me though. I had to close the Manager completely a few times (just because it has a tendency to "unsuspend" itself when I am using my PC), but there have never been any adverse effects before.

I have removed BOINC completely once again and installed the previous version (7.6.33).
Works like a charm now. Case closed, I reckon.