??? this still isn't fixed?
This is crazy.

When the problem is occurring, with Illustrator as the active application briefly press the following keys simultaneously to trigger a sysdiagnose:

Command + Option + Shift+ Control + Period (.)

Note: The sysdiagnose process can take 10 minutes to complete. Once finished, the folder “/private/var/tmp/“ should
appear automatically in the Finder and the sysdiagnose file there will look similar to this:
“sysdiagnose_2017.08.17_07-30-12-0700_10169.tar.gz”

send the sysdiagnose file that appears in /private/var/tmp/ to Adobe to pass along to Apple or send it to apple via https://feedbackassistant.apple.com and briefly describing the problem

Finished talking with the Adobe engineers that were working on my machine. They seem to have isolated the problem and they told me that they are working to get an update out as soon as possible.

Sam issue. Can't save. Can't Open. Can't do anything that involves retrieving a file from Finder.

Same issue. Lost a whole day of work. Not fun.

I'm having the same problem.

As I said earlier, I was speaking to a couple software engineers as they were tinkering with my machine remotely. They haven't isolated the problem yet but downloaded pages and pages of log files from my MacBook to analyze today. They will probably contact me tomorrow for another remote session. Whatever the issue is it's affecting most Creative Cloud applications in similar ways. Not a lot to share, I know, but I'll keep everyone posted as they progress.

"Also as a side note, suddenly, everything in Photoshop is a .PSD, regardless of what file format I save it in."

Per dnelson's comment, I also noticed this and included it in my email to ShareWithAI@adobe.com. It has the dubious consequence of saving over your PSD with a flattened jpeg, completely destroying your working file. Obviously that bug is not AI related (don't @me); however, as user Jennifer Moore pointed out below, both issues may actually be related to an issue with Finder and the enhanced security permissions that Apple introduced in Catalina. (I've noticed Google Chrome also crashing when trying to execute Finder related tasks, like uploading files from the desktop.) Guessing the engineers know all this, but I was kind of hoping for a fix by now. Saving files isn't exactly a new feature.

I am unable to save illustrator files in Catalina. This is a ridiculously critical issue that I cannot believe went unseen.

I CAN SAVE (command S) an existing file.

I CANNOT
1. Save As (menu or key commands)
2. Save as Template (“)
3. Export

I am running the latest version from the Creative Cloud. I have the latest Catalina PUBLIC release (No betas).

I am not running an Illustrator 3rd party plug-ins.

The ONLY WAY I can work is to duplicate an EXISTING AI file, work in it (often deleting everything first if I need to make a new file) and then saving it. I MUST change the name in the FINDER prior to opening it when I duplicate it as this is the only way to change the file name. As mentioned, Save As won’t work at all. NO dialogue box. NO crash. Nothing.

Also as a side note, suddenly, everything in Photoshop is a .PSD, regardless of what file format I save it in. I need to change there suffix in the finder after the save (ie; JPG).

I just don’t get how this critical stuff was all overlooked. I pay a premium for the subscription. What good is it if I can’t save my work?????

I just have initiated the backup through Timeline and restored to the older date into Mojave. That has saved me a lot of pain and after two days of tying, I had enough.

Same problem I'm experiencing and it's so upsetting. It's not easy to switch back to Mojave. Please help!!!

On the phone with Adobe right now. They have "remoted" into my machine and are working to diagnose the problem. As much of a pain this has bug been, I'm actually impressed that they reached out to me with a phone call. Hopefully this meeting yields an update fix for all of us.

This is crazy. Its been a week since people started encountering this and there still isn't a fix? This is incredibly damaging.

Upvote! Same issue.

Same problem here. This is critical. Can't use it.

I'm having the same problem.

I have the same problem. Won't save at all. I've opened and edited old files won't save, new files won't save either. Sucks.

Hello. I came across this post regarding issues with PS and AI with the new OS, Catalina. 
https://illustrator.uservoice.com/forums/601447-illustrator-bugs/suggestions/38772067-can-t-save-files-in-catalina
I am experiencing the same problem that I cannot save out files in Illustrator anymore. Whether interacting with the menu for save and save as. Nor with shortcuts. Nothing comes up. When I attempt to close the file, I do get this prompt. https://d.pr/i/ohlWKj But when I click on save, nothing happens. 
I am also experiencing a problem with Photoshop. When I Save for Web, I get this error message. https://d.pr/i/Yn0llU
When do you anticipate that these problems will be resolved? 

same here!
I can't save.

me too. losing work and hours of working time. I rely on illustrator for everything... and thus my clients do too. I've already missed one deadline. Apple support was no help (🙄). PLEASE FIX THIS.