What do you mean with 'turning off the backups'? Unchecking both checkboxes in the dialogue box showing up when clicking Edit in the 'Scheduled Copies' window?

And it now does so, ie, deleting the scheduled copies removes those two plist files. I only had discovered these two files relatively late in my debugging process (I was first only looking out for files with 'superduper' in their name). It is thus possible that I had set up a new scheduled copy (which didn't work), then deleted all files with 'superduper' in their name without first deleting the scheduled copies from within SuperDuper with therefore these two files remaining.

In short, I cannot say for sure that deleting the scheduled copies from within SD did not delete these two plist files. I only know that a new 'installation' of SD, encountering these two files failed to start upon mounting a destination volume (of a newly scheduled copy). And I know that only deleting the scheduled copies from within SD followed by deleting their folder in Application Support did not solve the problem.

But the next time this occurs, I will first check whether deleting the scheduled copies from within SD actually removes those two files or if them getting stuck might have caused that problem. (I might even be able to test this by booting of a clone from a few days ago.)

I mean deleting the scheduled copies, or unchecking the checkboxes, will delete those files. Since I originally told you to delete them, they were removed already, and are only back because you recreated the schedules. Why deleting them manually worked I don't know, since - again - they had already been removed.

Exactly same issue here.
Could someone help us, please ?

Thanks in advance.

See my replies to Manu - that will fix it.

I had not seen it.
I am reading it.
Thanks for quick and efficient support.

Solved ! :D

Great! Glad you're fixed.

Problem re-occured - nailed down part of the problem
 

The problem re-occurred (already some time ago but I just now came around to fix it).
In the end, it only started working again after I downloaded a fresh copy of SD and replaced the one in my /Applications folder with it. Before that I went through seven debugging rounds, always repeating all previous measures and adding one or two additional things. I cannot say that replacing SD in Applications was sufficient it but it was a necessary ingredient.

1) Deleted schedules, deleted 'Scheduled Copies' (in ~/Library/Application Support/SuperDuper!, restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

2) Deleted schedules, deleted 'SuperDuper!' (in ~/Library/Application Support), restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

3) Deleted schedules, deleted
- 'SuperDuper!' (in ~/Library/Application Support) and
- both 'com.shirtpocket.backuponmount' plist files (in ~/Library/LaunchAgents),
restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

4) Deleted schedules, deleted
- 'SuperDuper!' (in ~/Library/Application Support) and
- both 'com.shirtpocket.backuponmount' plist files (in ~/Library/LaunchAgents),
- 'blacey.SuperDuper!.savedState' (in ~/Library/ Saved Application State)
- 'SuperDuper!_xxx.plist' (in ~/Library/Application Support/CrashReporter)
restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

5) Deleted schedules, deleted
- 'SuperDuper!' (in ~/Library/Application Support) and
- both 'com.shirtpocket.backuponmount' plist files (in ~/Library/LaunchAgents),
- 'blacey.SuperDuper!.savedState' (in ~/Library/ Saved Application State)
(- 'SuperDuper!_xxx.plist' (in ~/Library/Application Support/CrashReporter))
- 'com.blacy.SuperDuper!' (in ~/Library/Caches)
restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

6) Deleted schedules, deleted
- 'SuperDuper!' (in ~/Library/Application Support) and
- both 'com.shirtpocket.backuponmount' plist files (in ~/Library/LaunchAgents),
- 'blacey.SuperDuper!.savedState' (in ~/Library/ Saved Application State)
(- 'SuperDuper!_xxx.plist' (in ~/Library/Application Support/CrashReporter))
- 'com.blacey.SuperDuper!' (in ~/Library/Caches)
- 'com.blacey.SuperDuper.xxx.plist' (in ~/Library/Preferences)
- app.com.blacey.Superduper!.playlist (in ~/private/var/db/BootCaches/xxx)
restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

7) Deleted schedules, deleted
- 'SuperDuper!' (in ~/Library/Application Support) and
- both 'com.shirtpocket.backuponmount' plist files (in ~/Library/LaunchAgents),
- 'blacey.SuperDuper!.savedState' (in ~/Library/ Saved Application State)
(- 'SuperDuper!_xxx.plist' (in ~/Library/Application Support/CrashReporter))
- 'com.blacey.SuperDuper!' (in ~/Library/Caches)
- 'com.blacey.SuperDuper.xxx.plist' (in ~/Library/Preferences)
- app.com.blacey.Superduper!.playlist (in ~/private/var/db/BootCaches/xxx)
- Replaced SuperDuper!.app in Applications with a newly downloaded copy
restart computer, re-create schedule, saved SD settings, quit SD, ejected disk, re-connected disk --> nothing happened

After step 6) I had to 'unlock' something (probably because I had deleted stuff in /private/var). And when it worked after step 7) I had to authorise a new application inside the SuperDuper! bundle to be run for the first time (because it was downloaded from the internet.

Next time the problem re-occurs, I'll start debugging from the other end (ie, start with replacing SuperDuper!.app and then add the boot caches and so on). If you are interested in the SuperDuper!.app bundled that needed replacing, I can retrieve it from a backup and send it to you.

Sounds more like something had happened to launch services and/or your script dictionary...

And running a new copy of SD for the first time resets something in launch services?

Yes, deleting your existing copy, then reinstalling is going to clear out Apple's cached information about the app.

Should the function 'Rebuild Launch Services Database' in Cocktail (under System -> Databases) do this as well or would it be the 'Clear Caches', System and/or User (under Files -> Caches)?

It should, although it may not re-fetched a cached script dictionary.