Software Update isn't down, but that's a bug in Big Sur that's been there since the beginning. You can restart and it might work, or you can try using this command in terminal to get Software Update unstuck:
Using the latest Intel-based Macbook Pro: 2020 13" 2.3 GHz / 32 GB / 2 TB. The command "sudo launchctl kickstart -k system/com.apple.softwareupdated" seemed to help. I can now see new updates.