Time Machine on MacBook shows 'No Available Time Machine Destinations'.

I have a partition on an external hard drive called "Sandisk Time Machine Backup Partition" and that's where the backups were being done until they suddenly stopped.

Does that drive appear in Disk Utility? Is it mounted? If so Time Machine shouldn't be giving you any trouble.

If it is mounted yet TM says it can't find it, then remove it from Time Machine's Settings, confirm when asked, and add it again. Doing that will not alter its existing backup history in any way.

Assuming TM is able to read that drive's contents it will tell you it already contains backups for that Mac, and if you want to continue using it or create a brand new backup. Don't create a new backup; choose the former and it will continue where it left off (after a long period of analysis... could take an hour or more).

Authenticate with your TM backup password when asked.

This backup drive is only used when I am on my MacBook and travelling, my backups when I am using my MacMini at home are onto a different external rive.

Unless I am misunderstanding something the two Macs are backing up to two separate backup drives, each one dedicated to its own Mac, and are completely unrelated to one another. That is the right way to do things.

Yes I was misunderstanding something. Dedicate external drives to TM. The entire drive, not a partition, subset, or share. It keeps things simple.

Unless I am misunderstanding something the two Macs are backing up to two separate backup drives, each one dedicated to its own Mac, and are completely unrelated to one another. That is the right way to do things.

Backups should be simple, if for no other reason it simplifies troubleshooting.

I understand. It's one of the reasons I recommend keeping things simple.

If you obtain an additional backup drive and add it to TM, then you can proceed with troubleshooting to include removing the problematic drive, erasing it, etc. It affords flexibility that you will need to fix whatever is wrong.

What I don't understand is that it worked fine before, then suddenly stopped (between 8 and 21 June this year), presumably following some sort of upgrade to OS.

Yes. At one point in the distant past Apple at least hinted at changes or improvements to TM as part of their release notes, but they don't do that any more. Apple routinely incorporates TM code changes along with minor macOS point updates, with only vague "improvements in reliability and performance" references. Such opacity makes it difficult to correlate changes in behavior.

When encountering unexpected TM behavior, again I endeavor to keep things simple. Methodically proceed from the innocuous (reboot) to the extreme (erase), but you never want to be without a backup, not even for a moment.

Leo Bloom (author)

Perhaps to have and use the spare USB rotational drives

on this occasion is best choice; when your primaries are

DOA & no longer functions? Dedicate 'non-partitioned'

entire external HDD of sufficient capacity for a long run.

I've several backup Time Machine capable Spares. While

only have four Macs here; two get used 8-10 hours a day.

[None are SSD, due to their unknown wear pattern cycles.]

.The Spares are still new & even so, might Fail on 1st run.

Be sure those chosen have practical format for TM B/UP.

Yes, it appears in Disk Utility and it is mounted. I don't know how to remove it from Time Machine's Settings since nothing appears in Time Machine settings, it doesn't even recognise the external drive, never mind the backup partition:

Unfortunately I can't do that since my home folder is also on the external drive. What I don't understand is that it worked fine before, then suddenly stopped (between 8 and 21 June this year), presumably following some sort of upgrade to OS.

Even having my home folder on the external drive is a nightmare these days - thanks Apple - oh! for the days when you could boot from an external.

So I guess I need another backup drive.

So I guess I need another backup drive.

Everyone does 😄