Great topic! For me:
KISS: 12/27/76 Rock and Roll All Over Tour - Cumberland County Area, Fayetteville, NC - my dad took me. It was my first ever concert.
Yes: 09/16/83 - Greensboro Coliseum, Greensboro, NC. Alan White had his limo pulled over at the gate after the show and signed some tour books, including mine.
Heart: 11/7/84 - Minges Coliseum, Greenville, NC
ZZ Top: 04/03/86 - Greensboro Coliseum, Greensboro, NC
Pink Floyd: 10/25/87 - Dean E. Smith Center, Chapel Hill, NC
Pink Floyd: 05/88 (I think) - Carter Finley Stadium, Raleigh NC. It had rained all day and 5 minutes before the show started the setting sun came out, a few airliners flew overheard and then the lasers started shining into the sky. It was so perfectly Pink Floyd, it was surreal.
Lynyrd Skynrd: 06/87, Greensboro Coliseum, Greensboro, NC. First tour since the plane crash and it was fantastic but emotional and almost heartbreaking. Alan Collins had to do his drunk driving speech as part of his plea agreement, I guess.
AC/DC with Cinderalla opening: 10/18/88 - Reynolds Coliseum, NC State, Raleigh NC. I was deaf for 3 days.
The Who: 7/27/89: Carter Finley Stadium, Raleigh, NC. All night long a bat kept flying around Entwistle's head and he swatted at it the whole show.
Man or Astroman?: 01/17/99 Cat's Cradle, Chapel Hill, NC. Without a single doubt one of the best shows I've ever seen in my life.
Blue Öyster Cult: 11/16/01 Lincoln Theater, Raleigh NC. This show stood out among the times I saw them because they played a TON of stuff they hadn't played in years.
Flogging Molly with the Rolling Blackouts and the Scotch Greens: 02/22/06 930 Club, Washington DC.
Gogol Bordello: 07/31/09 The Norva, Norfolk VA. THE ENERGY!
Asia: 08/10/10 Raleigh Memorial Auditoreum, Raleigh NC
Man or Astroman?: 11/06/10 Cat's Cradle, Chapel Hill NC (Yes, they are on my list twice)
Tour dates courtesy of my folder of ticket stubs...