Cool thread.
Many, many "bests," and I can't limit it to one, but I can cut it to 3. In chronological order:
Van Halen, 1984 Tour, Reunion Arena. Awesome band with their best lineup
The Cure -- 1986, Bronco Bowl, Dallas, Texas. First date with the girl I would marry. She drug me to it. I found, much to my surprise, that I liked the band. I also fell in love, so that prolly had a lot do with it. Can still smell the scent in the air when they played Charlotte Sometimes that night (no, not THAT smell...they had just opened one of the big doors behind the stage to let some air in when they kicked into that song, and a jasmine like smell permeated the place). It's a nice memory.
Stone Temple Pilots, Aragon Ballroom (or sumpin' like that), Chicago, 1993. I went up to Chicago on business and came home with a new favorite band. This is the only band I've ever "toured" with, following them from Dallas to New Orleans and on to Tampa the next year. Great fun. Those boys were really cookin' in those days, and Weiland wasn't too messed up yet (or at least not at that particular point in time).
Worst? meh, I block the bad ones from my memory.
Oh, most unexpected concert moment: Michael Jackson's big '84 tour with his brothers swung through Dallas at the same time the Van Halen boys were in town. As I recall, VH was playing multiple nights at Reunion (one of 'em being the show referenced above), and MJ and his brothers were playing multiple nights at Texas Stadium. I don't even recall which of the shows I saw first, but I think it was MJ one night and VH the next...was kind of working my way through a perpetual smoke filled haze back then

, so my memory may be a little off. One thing I definitely remember is it was hotter than hell -- summer concerts. Anyway, the VH boys apparently had a night off during all this and, ta-da, they appeared at the Jackson show. Walked through the audience on the concert floor (at least I think that's what happened...it's what I was told happened, but I was in the nosebleeds and wasn't alerted to what was going on until they were on stage...just heard the roar). Long story short, EVH played the solo on Beat It. Very unexpected and completely cool.