installed linux-image-5.13.0-39-generic package post-removal script subprocess returned error exit status 1

For diagnostic purposes please provide the full output of the commands (to be executed in a terminal window):

uname -a
lsb_release -crid
ls -l /boot/*vmlinuz*
sudo apt update
sudo apt full-upgrade

I'm having the exact same issue. Here is the data requested from my system:

Linux PaulsWorkstation 5.15.0-27-generic #28-Ubuntu SMP Thu Apr 14 04:55:28 UTC 2022 x86_64 x86_64 x86_64 GNU/Linux

Distributor ID: Ubuntu
Description: Ubuntu 22.04 LTS
Release: 22.04
Codename: jammy

lrwxrwxrwx 1 root root 25 Apr 28 17:41 /boot/vmlinuz -> vmlinuz-5.15.0-27-generic
-rw------- 1 root root 10246688 Mar 29 09:13 /boot/vmlinuz-5.13.0-40-generic
-rw------- 1 root root 11064224 Apr 13 23:47 /boot/vmlinuz-5.15.0-27-generic
lrwxrwxrwx 1 root root 25 Apr 28 17:41 /boot/vmlinuz.old -> vmlinuz-5.13.0-40-generic

Hit:1 http://us.archive.ubuntu.com/ubuntu jammy InRelease
Hit:2 http://archive.ubuntu.com/ubuntu jammy InRelease
Hit:3 http://us.archive.ubuntu.com/ubuntu jammy-updates InRelease
Hit:4 https://dl.google.com/linux/chrome/deb stable InRelease
Hit:5 http://us.archive.ubuntu.com/ubuntu jammy-backports InRelease
Hit:6 http://security.ubuntu.com/ubuntu jammy-security InRelease
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
1 package can be upgraded. Run 'apt list --upgradable' to see it.

Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
Calculating upgrade... Done
The following packages were automatically installed and are no longer required:
  hwinfo libhd21 libx86emu3 ruby-minitest ruby-power-assert ruby-test-unit
Use 'sudo apt autoremove' to remove them.
The following packages will be REMOVED:
  linux-image-5.13.0-39-generic
The following packages will be upgraded:
  base-files
1 upgraded, 0 newly installed, 1 to remove and 0 not upgraded.
1 not fully installed or removed.
Need to get 62.8 kB of archives.
After this operation, 10.3 MB disk space will be freed.
Do you want to continue? [Y/n] y
Get:1 http://us.archive.ubuntu.com/ubuntu jammy-updates/main amd64 base-files amd64 12ubuntu4.1 [62.8 kB]
Fetched 62.8 kB in 0s (173 kB/s)
(Reading database ... 248820 files and directories currently installed.)
Removing linux-image-5.13.0-39-generic (5.13.0-39.44) ...
/etc/kernel/postrm.d/initramfs-tools:
update-initramfs: Deleting /boot/initrd.img-5.13.0-39-generic
/etc/kernel/postrm.d/zz-update-grub:
Sourcing file `/etc/default/grub'
Sourcing file `/etc/default/grub.d/init-select.cfg'
Generating grub configuration file ...
using custom appearance settings
Found background image: /home/paul/Pictures/DJI_0484.JPG
Found linux image: /boot/vmlinuz-5.15.0-27-generic
Found initrd image: /boot/initrd.img-5.15.0-27-generic
Found linux image: /boot/vmlinuz-5.13.0-40-generic
Found initrd image: /boot/initrd.img-5.13.0-40-generic
/etc/grub.d/bin/grubcfg_proxy: error while loading shared libraries: libcrypto.s
o.1.1: cannot open shared object file: No such file or directory
run-parts: /etc/kernel/postrm.d/zz-update-grub exited with return code 127
dpkg: error processing package linux-image-5.13.0-39-generic (--remove):
 installed linux-image-5.13.0-39-generic package post-removal script subprocess
returned error exit status 1
dpkg: too many errors, stopping
Errors were encountered while processing:
 linux-image-5.13.0-39-generic
Processing was halted because there were too many errors.
E: Sub-process /usr/bin/dpkg returned an error code (1)

@Paul Briery: Please do not piggyback on somebody else's question, but create a question document of your own.
Even if the symptoms look similar, the cause can be different and may require different actions.

Your system has been broken by grub-customizer.
There are discussions in Bug #1969353 and Bug #1969793 about this problem.

Thank you so much for the links and the education on the etiquette of posting.

To Manfred Hampl regarding your first answer to my question posted 2022-04-29:

I ran your suggested commands in a terminal window. How do I copy the results for posting here?

There are different possibilities. In my opinion the easiest is:

In the terminal window click the right mouse button and in the pop-op that appears activate the display of the menu-bar
Then use the menu entries "edit - select all" and "edit - copy" for copying the output and paste it into the "Message:"-box of this question document

steve@GuardianIII:~$ uname -a
Linux GuardianIII 5.15.0-25-generic #25-Ubuntu SMP Wed Mar 30 15:54:22 UTC 2022 x86_64 x86_64 x86_64 GNU/Linux
steve@GuardianIII:~$ ls -l /boot/*vmlinuz*
lrwxrwxrwx 1 root root 25 Apr 28 10:08 /boot/vmlinuz -> vmlinuz-5.15.0-27-generic
-rw------- 1 root root 10246688 Mar 29 10:13 /boot/vmlinuz-5.13.0-40-generic
-rw------- 1 root root 11073600 Mar 30 11:29 /boot/vmlinuz-5.15.0-25-generic
-rw------- 1 root root 11064224 Apr 14 00:47 /boot/vmlinuz-5.15.0-27-generic
lrwxrwxrwx 1 root root 25 Apr 28 10:08 /boot/vmlinuz.old -> vmlinuz-5.15.0-25-generic
steve@GuardianIII:~$ sudo apt update
[sudo] password for steve:
Hit:1 http://us.archive.ubuntu.com/ubuntu jammy InRelease
Get:2 http://us.archive.ubuntu.com/ubuntu jammy-updates InRelease [109 kB]
Hit:3 http://us.archive.ubuntu.com/ubuntu jammy-backports InRelease
Get:4 http://security.ubuntu.com/ubuntu jammy-security InRelease [110 kB]
Get:5 http://us.archive.ubuntu.com/ubuntu jammy-updates/main amd64 c-n-f Metadata [1,148 B]
Get:6 http://us.archive.ubuntu.com/ubuntu jammy-updates/universe amd64 c-n-f Metadata [568 B]
Fetched 222 kB in 1s (272 kB/s)
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
2 packages can be upgraded. Run 'apt list --upgradable' to see them.
steve@GuardianIII:~$ sudo apt full-upgrade
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
Calculating upgrade... Done
The following packages will be REMOVED:
  linux-image-5.13.0-39-generic
The following packages will be upgraded:
  libinput-bin libinput10
2 upgraded, 0 newly installed, 1 to remove and 0 not upgraded.
2 not fully installed or removed.
2 standard security updates
Need to get 151 kB of archives.
After this operation, 10.3 MB disk space will be freed.
Do you want to continue? [Y/n] y
Get:1 http://us.archive.ubuntu.com/ubuntu jammy-updates/main amd64 libinput-bin amd64 1.20.0-1ubuntu0.1 [19.9 kB]
Get:2 http://us.archive.ubuntu.com/ubuntu jammy-updates/main amd64 libinput10 amd64 1.20.0-1ubuntu0.1 [131 kB]
Fetched 151 kB in 0s (518 kB/s)
(Reading database ... 205770 files and directories currently installed.)
Removing linux-image-5.13.0-39-generic (5.13.0-39.44) ...
/etc/kernel/postrm.d/initramfs-tools:
update-initramfs: Deleting /boot/initrd.img-5.13.0-39-generic
/etc/kernel/postrm.d/zz-update-grub:
Sourcing file `/etc/default/grub'
Sourcing file `/etc/default/grub.d/init-select.cfg'
Generating grub configuration file ...
/etc/grub.d/bin/grubcfg_proxy: error while loading shared libraries: libcrypto.s
o.1.1: cannot open shared object file: No such file or directory
run-parts: /etc/kernel/postrm.d/zz-update-grub exited with return code 127
dpkg: error processing package linux-image-5.13.0-39-generic (--remove):
 installed linux-image-5.13.0-39-generic package post-removal script subprocess
returned error exit status 1
dpkg: too many errors, stopping
Errors were encountered while processing:
 linux-image-5.13.0-39-generic
Processing was halted because there were too many errors.
E: Sub-process /usr/bin/dpkg returned an error code (1)
steve@GuardianIII:~$

Thank you.

Paul Briery's problem is the same as yours and I already told him:

Your system has been broken by grub-customizer.
There are discussions in Bug #1969353 and Bug #1969793 about this problem.

Well, while working this problem one of my drives decided to fail, the one with my Windows 10 installation on it. After I replaced the drive with a new one, I decided to completely wipe the remaining drive and do a clean installation of Jammy Jellyfish. Once I got Windows 10 installed on the other drive, I went in and added GRUB_DISABLE_OS_PROBER="false" to the appropriate file, ran 'update-grub' and rebooted my computer. It all works like a charm, both operating systems are available.

So, as afar as I'm concerned, my original problem is solved.