I'm not getting update notifications (since a couple of months) for gnome-software though available updates (e.g. for snaps and flatpaks) show in the application and automatic updates are enabled.
e.g. OS Updates like this one or nvidia driver updates or one of my snaps/flatpaks (audacity, godot, etc.)

I'm a bit lost at finding the right log to see what error keeps those notifications from appearing.
gnome-software 3.38.1 on Ubuntu 21.04
That's what I did find after a reboot:
Aug 30 07:51:15 system gnome-software[2066]: not handling error failed for action get-updates-historical: failed to build result for b8f11848dfcbf441df375b9f11b0e42d085f688b
Aug 30 07:51:16 system gnome-software[2066]: not handling error failed for action refresh: E: Failed to fetch http://de.archive.ubuntu.com/ubuntu/dists/hirsute/InRelease
Aug 30 07:51:17 system gnome-software[2066]: CSS parse error 1:107: Using one color stop with linear-gradient() is deprecated.
Aug 30 07:51:17 system snap-store[2120]: not handling error failed for action get-updates-historical: failed to build result for b8f11848dfcbf441df375b9f11b0e42d085f688b