When I click on "check for software updates" it just spins and spins and doesn't stop.

There are factors:

1 - Like poor Wifi connection

2 - Slow wifi connection

3 - Too many other people downloading at the same time and Apple Severs are overloaded

4 - Software on the computer interfering with the download process including AntiVirus Software

Notation: Disabling the AV Software may or may not solve the issue. Removal of the AV Software may increase the chances of the download to perform as expected.

5 - Running behind a VPN which may increase Packet losses or corruption of the download 

6 - If via wifi too many Other devices using the wifi at the same time reducing the Bandwidth to your computer. 

7 - Best Results are via Ethernet Cable to Router

8 - My Favourite >> Restart in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal

Try Again using the Software Update Applet in System Preferences

9 - Worst case - Restart in Recovery Mode and choose Reinstall macOS. This in theory will pull the latest version of Big Sur available and commence the installation

I understand Software Update is "Checking for updates..." with no end in sight.

Several workarounds have been proposed, such as restarting your Mac in "Safe Mode" and checking for updates while in that mode, and / or resetting NVRAM. Try both.

How to use safe mode on your Mac - Apple Support

Reset NVRAM on your Mac - Apple Support

Related: Software Update stuck at "Checking for updates"… - Apple Community