Laptop: Thinkpad E485, 32GB RAM, Ryzen 3 2200U with vega graphics. Kernel 5.3.0-53-generic
Symptoms: Fails to shut down. Both from gui and shutdown -h now
results in black screen, unresponsive keyboard, but the power light stays on. The laptop never truly shuts down, and it results in battery being discharged to 0%.

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4 Answers
A bug has been filled as #1880041
Please pass by and click "It affects me too" so it gets more attention

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The issue passed after kernel upgrade to 5.6.14 using the ukuu
tool:
I had problems installing ukuu
from the provided PPA, so I had to build it from source. I used the instructions posted on the ukuu git repository page.
sudo apt-get install libgee-0.8-dev libjson-glib-dev libvte-2.91-dev valac
git clone https://github.com/teejee2008/ukuu.git
cd ukuu
make all
This step produces a lot of warnings, or at least it did on my laptop, but those are not really important, the resulting binary worked fine.
sudo make install
After the last step just run the ukuu-gtk
binary and install selected kernel. I chose 5.6.14, because it was the latest one at the time of writing this answer. The shutdown problems went away, there may be some incompatibilities, but you have to test it for yourself.

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2020-06-18 edit
Kernel 5.3.0-59 fixes the issue. I strongly recommend you to update your install and only use kernels from official repositories rather than ones suggested by others that were probably never tested with the rest of your software and won't get any updates.
Original answer
Have you tried booting with an older version of the kernel? See this answer for instructions on how to do so.
If no other version than 5.3.0-53 is installed, try installing 5.3.0-51 (it does solve the issue on my computer)
sudo apt install linux-image-5.3.0-51-generic linux-headers-5.3.0-51-generic
Until you upgrade to 20.04 or 18.04.5 ships with kernel 5.4 (and assuming it does fix the issue), I advice you to stick with 5.3.0-51.
In order to force your computer to boot on this kernel version, see this answer from the same thread.
Obviously, if a new version of kernel 5.3 gets released, try if it fixes the issue and then set GRUB_DEFAULT=0
to boot on the newest kernel.
Hopes it helps you!

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I had the same problem on T495 AMD Ryzen 7 PRO 3700U. I manage to solve it on 5.3.0-53 by installing AMD drivers from here https://www.amd.com/en/support/kb/release-notes/rn-amdgpu-unified-linux-20-10
I hope this solves the issue for you as well :)