Why will the software update bubble not go away??

Open the System Preferences. At the top right is where you'll likely see a red dot, urging you to complete signing into iCloud. If you don't use iCloud (I don't), you can instead click the sign out button. That will remove the "urgent" dot.

Open Software Update. It may say the OS is up to date, but you may see a secondary button about the middle of the panel, above the Advanced button. That's where Apple sticks non-OS updates, such as those for Safari, or security updates.

robsall wrote:

That file doesn't exist :-)

Yes it does. While in the Finder, press "command shift G", copy then paste ~/Library/Preferences/com.apple.preferences.softwareupdate.plist into the window that opens. Return. The file will highlight. Move it to the trash.

Huh. I thought it worked with anything text-based.

Brave works. BBEdit, Notes, Pages... can't find anything in which it doesn't work.

Firefox wants to intercept the secondary-click. There's probably a way to customize it though (as there usually is with FF).

Delete this file then restart.

~/Library/Preferences/com.apple.preferences.softwareupdate.plist

robsall wrote:

That file doesn't exist :-)

Easy way:

Another possible solution: Red badge on SP

This article might help: Ignore Catalina Upgrade Prompt in Software Update

John Galt wrote:

Easy way:

Yes, unless you use Firefox. :)

Firefox has no "Look Up" integration. It also has no text replacement integration. I still like it though. :)

Yep.