I think my favourite moment was when I saw them in 2013 and they were playing in a big stadium. I'd seen them once before in 2010 but I wasn't a fan back then (I went because my friend wanted company and I thought they seemed like a cool band).
Me and my friends were in front row and we were having so much fun and usually at gigs even though I'm enjoying it, I'm not necessarily thinking "oh I never want this to end this is so great". But during that gig I think I even yelled out loud something like THIS IS THE BEST DAY OF MY LIFE etc etc and god. I've never had so much fun than during that gig.
I personally thought Madness was a terrible song but it had sort of become an inside joke between my friends like for example if it came on the radio while we were in a car we'd sing along over dramatically. And I remember yelling mA-MA-MA-MAD-MAD-MAD with my best friend very very loudly and people looking at us like "...okay calm down" and us laughing about it and I just had so much fun I'm never getting over that stupid gig.
I also touched all of them since they ran past where we were! I got to hold Matt's hand for a brief time during Undisclosed Desires so that was pretty amazing...
So yeah. I probably have lots of favourite Muse member moments from videos and etc but that gig was the best thing that's ever happened to me. :')