How to get Wine working on Catalina?
I was able to build Wine 64 bit from source on macOS Catalina and successfully run Notepad++ 64 bit on it.
My steps were more or less something like that:

Download Wine 4.20 from https://dl.winehq.org/wine/source/4.x/wine-4.20.tar.xz
Extract it
In terminal go to extracted directory
Run: ./configure --enable-win64
It showed that it couldn't find bison, so I installed it with macports (sudo port install bison).
After I was able to run ./configure --enable-win64 successfully without errors in terminal run: make
It compiled after more than 30 minutes.
I run it with: ./wine start
It opened windows console and here I was able to cd to Notepad++ 64 bit directory (downloaded from https://notepad-plus-plus.org/repository/7.x/7.0/npp.7.bin.x64.zip note that newest version did not work and threw some errors about some dll's)
I typed notepad++ to run exe file
It showed some errors about freetype fonts so I installed them with macports and I had to copy them from /opt/X11/lib to /usr/local/lib before wine detected them
After fixing freetype fonts problem I was able to run notepad++
Also, you might want to run: sudo spctl --master-disable to disable gatekeeper if you will see some system alerts about loading app from an unknown developer.
Also, note that instead of ./wine start you can run ./wine explorer for graphical file manager instead of console

Those steps might not be exactly precise, as I'm not sure if I did not miss something but hope it will help someone.
Also, note that obviously wine on macOS Catalina will be able to run only some 64-bit windows apps and all 32-bit windows app will not work. I also tested it with IrfanView 64 bit (https://www.fosshub.com/IrfanView.html?dwl=iview453_x64_setup.exe) and it also works fine.
In case you wanted to get wine working for purpose of running 32-bit games then unfortunately it will not work with wine but you can try with VirtualBox. You can check https://www.youtube.com/watch?v=3AO8dF0vviQ for some performance improvement tips (in brief: using VM with retina disabled, VboxVGA setup, disabled mouse integration, limiting FPS with RivaTuner).
You can also try commercial CrossOver software by CodeWeavers which provides wine that supports 32 bit apps on Catalina. You can download the trial version for free and test it for free for 14 days. Also probably 32 bit apps will be supported by a normal free version of wine at some point.
EDIT:
Since July 2020 PlayOnMac (https://www.playonmac.com/en/) works again on macOS Catalina so you can download it and run windows apps with it which I think is much simpler. Good news is that it also supports running 32bit windows apps on macOS Catalina. Installing apps with it is a little more complicated than with commercial CrossOver but it is free and worked with 2 apps which I tested. I checked it with Notepad++ 32bit and with 32bit game Tomb Raider Anniversary Demo and I was able to get both apps working on macOS Catalina. Also I was able to install RivaTuner 7.2.3 app (with dotnet35 winetrick and manual installation of Visual C++ 2008 Redistributable package) to limit FPS in game which I recommend if someone wants to limit cpu and cpu fan usage.