The 1st generation M1 based Mini most certainly has graphics "quirks".
For me, the quick version is some monitors work, some don't. It's most likely going to be a software fix though. I've been lucky that over HDMI I've tried 2 monitors, one 1920x1080 and one 3440x1440 and both worked flawlessly.
Over USB-C I've had a USB-C to DisplayPort 1920x1080 monitor work fine. A older 1280x1024 USB-C to DVI monitor from LG not work (green static everywhere). Then (using the same cable) tried another older HP DVI 1280x1024 and it worked fine. Swapping the cable back to the LG, it was back to green static. Move that LG (again, same cable) over to test it on a 2017 iMac and a 2015 13" MacBook Pro, and it works fine on both of those. Back to the new Mac mini, green static. Old Dell 24" 1920x1200 DVI monitor works fine also using that same USB-C to DVI cable on the mini. So I gave up on using the LG as a second monitor and just used the older HP instead.
Current setup, No-name HDMI 1080p primary monitor, old HP 1280x1024 as USB-C to DVI secondary monitor, working fine, no graphic glitches on screen, but waking from sleep takes about 30 seconds for the displays to settle in sometimes. They power/sleep/power a few times before finally turning on for good. Once they do everything is fine.
/edit. In case someone is searching online to figure out why their LG display won't work properly with the M1 mini, the monitor I was having issues with was a
LG Flatron L1920P. We have a few laying around here and none worked on the M1 Mini using DVI. Didn't try VGA. It's a super old (like 16 years) monitor, so probably won't be a issue for most. But I can say with certainty that as of Jan 15th, 2021, this display is not compatible with the M1 mini. (Might work later on with a software update, who knows)