1) Which external monitor are you connecting to? What is the model number?
2) What sort of USB-C cable are you using? Are you using the cable that comes with the monitor or are you using just any USB-C cable?
3) Have you tested the USB-C cable and monitor on another device? Does it work through there?

I have tested a work unit via USB-C with my BenQ EX2780Q monitor as well as with my Macbook Pro M1 and Macbook Air M1 and it all works fine.

A few problems encountered so far. Not sure Big Sur or M1 problems.

1. Video would play with a black screen. VLC+mp4 or mkv. VLC with mpeg2 in a DVD folder. Twice so far. Had to restart.
2. Safari losing Command-~ shortcut for switching windows. Quitting doesn't help. Had to restart. Once or twice so far.
3. Command-Tab app switcher would sometimes get stuck in my 2nd monitor and refuses to come back to my main monitor. Then after awhile, it does. Psychopathic Neural Engine at work. Happened a few times but not that often to be annoying.
Solved, OSI Layer 8 error. The App Switcher follows the monitor where I last accessed the dock.
4. Telco phone calls via my iPhone seems to be a lot less reliable than my MBPR Late 2013. Keeps disconnecting or losing audio or mic. Happens half the time. Enough for me not to want to use the Mac to make calls.
5. Screenshots stops working after awhile. ⌘-⇧-4 activates the selection pointer but doesn't capture anything. ⌘-⇧-5 brings up the screenshot bar but is unresponsive to anything other than Escape.
6. Illustrator 23.1.1 beachballs a lot when switching windows.

The only thing special that I have that may be different from you guys is that I have a 3rd party kext running, Tuxera NTFS. To install this, I had to change my Mac's Security Policy to Reduced Security.

And I have one System Extension running, Little Snitch.

Are you using Bettertouchtool? I suspect that may be conflicting with screenshot capture. Will test it out next time the problem occurs.