20, wow!! I would have loved to see them properly back in the AITB and COT days, I was a bit too young then though. *sniff*
Here's my lot.
1. 27/11/2003 - Wembley Arena (supporting Muse on their Absolution tour)
2. 07/12/2003 - Exeter Westpoint (supporting Muse on the same tour)
3. 21/02/2006 - London Astoria
4. 12/02/2008 - Porchester Hall
5. 15/04/2008 - Brixton Academy
6. 14/06/2008 - Delamere Forest (Mantraluna!!)
7. 11/10/2008 - London Roundhouse
8. 12/10/2008 - London Roundhouse
9. 13/10/2008 - London Roundhouse
10. 23/10/2008 - Manchester Apollo
11. 30/10/2008 - Manchester Apollo
12. 01/03/2009 - Southend Cliffs Pavillion
13. 07/03/2009 - Blackpool Empress Ballroom
14. 14/03/2009 - Wembley Arena (headlining this time!)
15. 08/07/2009 - Bridgewater Hall + Halle
16. 09/07/2009 - Bridgewater Hall + Halle
So now that I've actually counted them up I'm not that far off 20...!
I like telling the story that I first discovered both Elbow and Muse on the same compilation CD series, Reloaded, back in 2001. It was a sort of indie/rock version of the pop chart compilation albums, though there were only about 4 of them in the end. Through the Reloaded series I fell in love with Red by Elbow and also fell in love with Muse as several Muse songs were on there.
Elbow and Muse have been my two favourite bands since then, so you can imagine how perfect it waas that my first ever gig was no.1 on that list above, Muse supported by Elbow!! And since Elbow were the support, and the very first song they played was Red, that makes Red the first song I ever heard live, and Elbow the first band I ever saw live. That was particularly special for me since Red was the first Elbow song I discovered and it saw me through some pretty rough patches.
This might also help explain why several people have seen me toting a Muse bag at many of the Elbow gigs... as soon as Elbow merch release a bag then I'll use that instead.

I think I've rambled on enough here!