There is a FAQ on the Wolfram site that Mathematica 12.1.1 and earlier are not supported on macOS 11.
FAQ for macOS 11
Here's the FAQ:
Will my Wolfram desktop product be compatible with macOS 11 Big Sur?
Wolfram software designed for earlier versions of macOS may not
function on macOS 11. Wolfram Research cannot guarantee that any
previous versions of our products will be compatible with the released
version of macOS 11.
We are committed to ensuring that Wolfram product versions released
after macOS 11 will be fully compatible with macOS 11.
To ensure that you will be able to use your Wolfram product, please
use a supported operating system.
We will continue to update this page as further information becomes
available.
As I recall (and I have been using Mathematica on Macs since version 1.1), this is the first time that there has ever been such a disconnect.
** NEW INFORMATION **
After installing the final release version of macOS 11 (Big Sur), I find no problems at all with Mathematica 12.1.1. I do not have earlier versions installed. The issue noted below only shows up in the Big Sur Betas (and not at all for Mathematica 12.2 beta).
[Obsolete information]
Having said that, I have been testing 12.1.1 and the 12.2 beta under Big Sur. The only issue I have found with 12.1.1 is that Apple has inserted a menu item in the help menu (of all apps, not just Mathematica) : "Send Feedback to Apple". In every other app I have running on Big Sur, this causes no problems at all, but with Mathematica 12.1.1, it messes up the menu, effectively pushing the response 'down one'. So, I you select "Enter Activation Key", you get the dialog for "Why the beep", etc.
The problems that this causes can be worked around. For example, to get the documentation center, I had to enter a function in a notebook, and then use the pop-up menu to get to the documentation center. Once it is opened, it works just fine.
The 12.2 beta fixes this problem. Since I am in the beta test program, I have been using the beta for both Catalina and Big Sur. No real issues to report with the beta, but I have not tested all the new functionality extensively.
I have reported this issue to support and pre-release. Since every other app I have, including large cross platform third-party apps like Finale, do not have this issue, it seems to me that something was missed in moving the front-end from Carbon (32-bit) to Cocoa (64-bit).
[End obsolete information]
As to whether it is safe to upgrade to Big Sur, the answer is, it depends. It is always possible that there are issues with 12.1.1 and macOS 11 that I did not see, but other than the menu issue, I have not seen them. If you can live with the wonkiness of the Help menu, there should be no down side.
For myself, I already have the 12.2 beta, so I will be upgrading my main computer to macOS 11 tomorrow. The new nifties in Big Sur are very nice indeed. I also have Premier service, so I will be getting the release version of 12.2 when it is released "free".
For others who have premier service, upgrading should not be a problem in most cases. However, the lack of support for older versions may be a real problem for people who have to pay for every upgrade, or who need an older version for some reason.
I watched the Apple keynote yesterday, and I will be getting a new Apple Silicon MacBook Pro as soon as the 16 inch replacement is released. I think that a lot of other Mathematica users will be upgrading -- and in a year or so, only Apple Silicon Macs will be sold. This means that anyone using Mathematica older than version 12.2 will be forced to upgrade.
At the Virtual Wolfram Tech conference, they announced that Mathematica 12.2 would (probably) be released in November. The timing could not be a coincidence. It is my guess that Wolfram intended to release 12.2 once macOS 11 is available, which turns out to be tomorrow. For all I know, there may be some special upgrade path that will smooth the transition for most people. Time will tell.
So, To answer your question: If you are adventurous, it should be ok to upgrade to macOS 11. If not, you will need to wait for the release of Mathematica 12.2 before installing macOS 11.