Spammers are sending thousands of emails which show my domain as the
sender. I'm now getting three thousand "Undeliverable", "Delivery
Failure", "Failure Notice", "Delivery Status Notice (Failure)",
"Returned mail: see transcript for detail" emails each day, and the
problem is getting worse. My question is how can I stop whoever is
using my domain (email address) from doing so?

I don't know what the definitive solution may be, but first of all you
should report this directly to your service provider. They may be able
to patch up the hole.

Not much.  If there was an easy answer to this, the global war on spam
would take a major step forward.  With so much spam being sent from
"zombie" networks and/or overseas, the likelihood of finding a
prosecutable offender is slim though.  Best to try to set up some spam
filters to get rid of the false returns.

The same has happened to me.  After considerable research I found that
what is happening is the spammer is using fake addresses from your
domain as return address for his spam emails.  The catch all account
receives all bounced emails which is what you are getting.  Apparently
there is no way to prevent people from using any return address they
like in their outbound emails.  I simply set the 'catch all' account
to an empty account and set up a autoresponder to send back 'A spammer
is illegally abusing this domain for return addresses.  Please email
xxxx [AT] yyyy.com if you have sent a real email to this account and
are receiving this message.

Of course I made xxxx [AT] yyyy.com another forwarding account to my
real account so I could always cut it off if the spammer caught on.

Good luck

Yes, someone has been using my Email address as a return address, and
I'm also getting 'undeliverable' responses.

Annoying, but not much I can do about it.

I use Mailwasher to screen my mail.