I saw:
The The on the “Dusk” tour
Genesis in the “And Then There Were Three” tour
Phil Collins on the “Hello I Must Be Going” tour
Peter Gabriel on the “Plays Live” tour
Talking Heads on the “Stop Making Sense” tour
King Crimson on the “Beat” tour
U2 on the “Joshua Tree” tour
Pearl Jam on the “Ten” tour
Mike Oldfield on the “Tubular Bells” tour
Just a few tour names I remember of the top of my head:
Duran Duran - Sing Blue Silver Tour (Los Angeles - 1984)
Depeche Mode - Music for the Masses Tour (Pasadena - 1988)
Erasure - Phantasmagorical Your (San Francisco - 1992)
Eagles - Hell Freezes Over Tour (Yokohama - 1995)
Pet Shop Boys - Nightlife Tour (Tokyo - 2000)
Beyonce - Beyonce I Am… Tour (Tokyo - 2009)
Lady Gaga - Monster Ball Tour (Yokohama - 2010)
I saw 8 shows on that final North American tour, so I guess I saw The Who on their moejoe tour.
U2 - War
The Police - Synchronicity
Talking Heads - Stop Making Sense
Laurie Anderson - Mr. Heartbreak
Lou Reed - Magic and Loss
The Eagles - Hell Freezes Over (or it might have been the tour after that)
Bright Eyes - Digital Ashes in a Digital Urn
Peter Gabriel - Us
Those are the ones I remember off the top of my head.
Where did you see Stop Making Sense? What a terrific show. I was in college and saw them at the newly opened Hult Center in Eugene Oregon.
I saw David Bowie on the Glass Spider tour, summer of 1988. I saw him in Veterans’ Stadium in Philadelphia, on the second night of the US tour. Peter Frampton was playing guitar in his band, Squeeze opened for Bowie, and Tommy Conwell and The Young Rumblers opened for Squeeze. Good show.
Rock of the Westies came out in October 1975. 
Saw Paul McCartney (and his unnamed band) on the Tripping the Live Fantastic tour. It was pretty great.
I sit corrected. Sept. 29, 1975 San Diego Sports Arena. I was still with my high school girlfriend, so I estimated a little early in the relationship.
Chuck still trashes me on Bruce.
Me too – in San Diego. Where did you see them?
I saw Rush on the **Moving Pictures tour -
I saw Yes on the Drama **tour - in the round
I saw Genesis on the Duke tour
I saw **Pat Travers on the Black Pearl **tour
No worries, I’m still jealous. 
Def Leppard:
Seven Day Weekend and Never Ending Weekend Tours [Pittsburgh , PA, Wheeling, WV, Cincinnati, OH, Charleston, WV]
Slang Tour [Pittsburgh,PA]
Euphoria Tour [Huntington, WV]
X Tour [Huntington, WV and Columbus, OH]
25 Year Annaversary Tour [Baseball Park Tour]-Charleston, WV and Columbus, OH
YEAH! Tour [Columbus, OH, Ottawa, ON, Sarnia Bay, ON, Detroit, MI, Indianapolis, IN, Cincinnati, OH, Red Rocks, CO]
The Sparkle Lounge Tour [Columbus, OH, NY State Fair, Syracuse, NY. Would have seen them in Winnipeg ON and Ottawa, ON, but the shows were cancelled due to the band being sick]
KISS:
**Reunion Tour-**Charleston, WV
Psycho Circus Tour-Charleston, WV
Metallica:
St Anger Tour-Columbus, OH
Rush:
Snakes ‘n’ Arrows/Moving Pictures Annaversary Tour-Columbus, OH
Duran Duran:
**Seven and the Ragged Tiger Tour **[Pittsburgh, PA]
Notorious Tour [Pittsburgh, PA]
**Big Thing Tour **[Pittsburgh, PA]
INXS:
Kick Tour [Pittsburgh, PA]
Queensryche:
**Building Empires Tour **[Charleston, WV]
Should I tell a story about seeing The Quarrymen?
Naaah…
Ozzy Osbourne on The Ultimate Sin (with opening act Metallica)
Areosmith on Permanent Vacation
U2 on Zoo TV
Van Halen on 1984
Bon Jovi on Slippery When Wet
Iron Maiden on 7th Tour of a 7th Tour
Scorpions on Savage Amusement
Haven’t gone to a live show in about 20 years…
I saw Simon and Garfunkel on the tour they did after the Central Park reunion concert; don’t know if the tour had a name, though.
Was that the one that was filmed for her concert film, Home of the Brave?
I don’t know about anybody else, but I grew up in the 70/80s, era of arena tours and I literally saw at least 5-10 name band concerts a year between Rochester, Toronto, Buffalo and Syracuse, and an occasional one in Boston, NY, Cleveland and Chicago in the US, and Paris France. Hell, I saw Rush back in the early 70s when they were still an opening act, and my first concert ever was Black Sabbath back in 1969.
Which one?
Seriously, though, a few from my own experiences;
I saw R.E.M. on what was probably the Reckoning Tour in 1984 - probably my best chance to claim seeing a huge band before they were huge. Third row, I think it cost me $11.
KISS on the Dynasty tour.
Judas Priest on the Hell Bent For Leather tour (opening for KISS on the Dynasty tour).
Also saw KISS on the Animalize, Hot In The Shade, Revenge, Alive Worldwide and Sonic Boom tours.
Van Halen on the 1984 and 5150 tours.
AC/DC on the Who Made Who, Blow Up Your Video and Ballbreaker tours.
ZZ Top on the Afterburner tour.
Deep Purple on the Perfect Strangers tour.
Bon Jovi (opening for Ratt) on the last date of the 7800° Fahrenheit tour, the last gig they played before recording Slippery When Wet.
Black Sabbath on the Forbidden tour, saw the last gig Cozy Powell ever played with them.
Mötley Crüe on the Theatre Of Pain tour.
Ozzy on the Bark At The Moon and No Rest For The Wicked tours.
Too many Motörhead gigs to keep straight.