Out of the many many gigs I been to over the years, these are the ones that burned the brightest impression in my mind for being something more than just “live music”…
Jesus Lizard – incendiary hardcore gods (I wish I had got chance to see Big Black though) – saw these at a little, now sadly demolished, club in Manchester (UK) called The Boardwalk.
The Fall – many many times,many many different venues, and never once have they ever disappointed.
Galaxie 500 – spectral aural swoonyness. At the now sadly demolished International club.
Yo La Tengo – fantastic! Saw these the last time they played here, at the University, and when they played again, later the same year in Liverpool (I loved the previous gig so much I drove all the way there to see them again)
808 State (supporting Happy Mondays at Manchester G-mex many years ago – totally and completely blew them off stage – then saw them again at the Manchester .v. Cancer gig at the M.E.N Arena back in January, and again, they were just incredible… there should be a DVD of this gig coming out soon, and if they show the audience, you just might catch nemesis dancing his happy dance
).
New Order – at the aforementioned Manchester ‘v’ Cancer gig – I have seen them live before, and on record they are one of my fave bands, but never did it for me live… until January this year at that gig… they came on, and Bernard gave it “Hi, we are New Order, and we are playing Joy Division”, and they went straight into Transmission, and I was just like… WOW… and indeed, they just played Joy Division songs. And they were stunning. And I was happy.
Jon Spencer Blues Explosion. Coruscating and … err… explosive. 
The Chills – these were just incredible, Martin Philips has written some of the most wonderful pop tunes EVER. And live…they were even better!
The Chameleons – another group who I think wrote some of the most wonderful songs ever… and again, live, songs that on vinyl \ CD knocked me out suddenly took on whole new dimensions live.
Stereolab – I’m running low on hyperbole… let’s just say the were mesmericaly hypnotically awesome.
Maria Mckee - Saw her live with Loine Justice back in… 87 I think it was… and also seen her live solo a few times too – knocks me dead without fail.
Jeff Buckley – I liked Grace, but was not like a massive fan etc… but still, I went to see him live… and after the gig, I could barely speak (which is a good thing). 
I notice in the thread that some have seen some groups \ artists I would love to see, but never managed to… so far (and in some cases never will see due to splits \ deaths etc)… the main ones being:
Bon Scott era AC/DC, Led Zep, The Pixies, Big Black, Smashing Pumpkins, Cat Power, Beastie Boys, Tom Waits