All of these have 5 star recordings. To me these are probably the four most essential bootlegs. After these, I recommend going on a voyage of discovery in the catalogue!
My personal choices would be: 1980-10-14 - Hilversum, Netherlands - KRO Studios 1981-11-22 - New York, New York - The Ritz 1983-05-30 - Devore, California - Glen Helen Regional Park 1984-11-21 - Dortmund, Germany - Westfalenhalle 1987-04-29 - Rosemont, Illinois - Rosemont Horizon 1987-11-07 - Denver, Colorado - McNichols Sports Arena 1989-12-26 / 27 / 30 / 31 - Dublin, Ireland - Point Depot 1990-01-05 / 06 / 09 / 10- Rotterdam, Netherlands - Sportpaleis Ahoy
1992-04-23 - Vancouver, British Columbia - P.N.E. Coliseum 1992-06-11 - Stockholm, Sweden - Globen 1993-08-28 - Dublin, Ireland - R.D.S. Arena 1997-08-28 - Leeds, England - Roundhay Park 1997-09-23 - Sarajevo, Bosnia and Herzegovina - Kosevo Stadium
2001-10-10 - South Bend, Indiana - Joyce Center 2001-10-25 - New York, New York - Madison Square Garden 2006-12-04 - Saitama, Japan - Saitama Super Arena
There's many more great performances out there. But for now, those should serve you well. I'm sure other people will come in and add to my list (I think I actually included all that Mr_Trek suggested, so definitely check those out). I've not included many from the 2000's, and none from 360 (I'm running tight on time!). But as Mr_Trek says, check these out then go and explore
Welcome! Oh man, how much time you have? I can go on and on about shows everyone must have, but the lists Olof and Tim gave you should serve you as a starting point. The best thing is when one show takes you to another unexpected gem. Don't rely on 5 stars ranked shows only, there are a lot of 3.5, 4 star recordings that sound very good. I found the reviews here very useful for choosing a good bootleg.
Let us know of your discoveries!
Oh, and I couldn't resist: 1983-03-24 Glasgow, 1984-11-06, 1984-11-07 Glasgow, 1987-07-08 Brussels, 1989-12-11 Paris (Paris Love Town), 1998-02-11 Santiago and 2001-10-27 New York must be in everyone's collection, among many others.
My recomendations, one show per tour, I'll go with these ones:
Early Days:
Dublin National Stadium.
Best show of these days, lots of non-album tracks, decent sound.
It's also a meaningful show, Island executives were present and after this show they signed the band to their first contract.
Boy Tour
Paradise at Boston (1st show).
They played 2 gigs that day, I like the first one better, with the combo Ocean/11 O'clock both at the beginning and at the end of the show.
October Tour
St Patrick's Day at the Ritz in NY.
Many people will prefer Lido Beach (which would be my 2nd choice), but I'd go with this one.
It's St Patrick's Day, in New York, the sound is good, and this is one of only 3 shows of that tour when they played Tomorrow.
War Tour
Edinburgh on the first leg.
Here the obvious choice would be Red Rocks, but you won't find a bootleg in here (there's this thing called YouTube, though...).
Edinburgh has a great sound, great crowd, and great show.
Also, there's the first show of the tour, Dundee, the one and only time they played Like a Song, but the sound is not the best out there.
Unforgettable Fire Tour
Milan.
Great show, great sound. The next day show (Bologna) is also great, but Milano has one of only 6 performances of Indian Summer Sky, so I'll pick this one.
Alternatively, Edinburgh and Glasgow are great shows earlier on the tour.
Joshua Tree Tour
Austin
The sound and the show is great. Many people prefer the Rosemont show, but I find the performance way better in the Austin show.
Alternatively, the McNichol's Arena in Denver, nov 7th is great, but if you don't mind too much if the sound is not really great, then you have to give it a shot at McNichol's Arena in Denver, Nov 8th, which is the show recorded for the Rattle and Hum movie (including the "fuck the revolution" Sunday Bloody Sunday and almost everything in Black and White of the movie).
Lovetown Tour
Anything from the Point Depot shows in Dublin at the end of the year or those in Rotterdam in 1990.
Everybody seems to have a favourite between them, the obvious choice for me would be the new year's day concert, but it's not available in here.
If you don't mind the sound too much, Sydney Nov 18 is complete and has one of the few Hawkmoon 269 performances.
Zoo TV
Stockholm
Keep in mind that for this show, MTV had a contest and the winners appeared in the screens at times during the show, and they're a bit annoying with their screaming, but you can get over it and enjoy a great show still.
Also, half of ABBA joins the band for Dancing Queen.
Rotterdam, Gent, Dortmund, Tacoma and many many shows are great from this tour...
Zoo TV -Zooropa/Zoomerang-
Wembley, Aug 12th
Tough choice between this one, Leeds the next day, and the last show in Dublin, all three great shows with great sound (and I'm not even counting Sydney official release at the end of the tour).
But this show has...
-Sarajevo satellite link up
-(One of) The best Bad performance, with violinist Jo Shankar joining them on stage
-(One of) the best Bullet performance, with Bono ranting totally pissed off at the big nations because of the Bosnia conflict.
-One of only 3 performances of Zooropa (before 360) and 7 of Babyface.
After this one, get the last show in Dublin, which is also great, and then there's the Sydney show but that one's not here, so...
Popmart Tour
Sarajevo
Because of all the meaning behind this show. Bono has no voice, which makes it either a "bad performance from the band" or a "incredibly emotional performance from the band". I think it's the latter.
Alternatively, there's Leeds, which also gets lots of votes.
Elevation Tour
Notre Dame (South Bend)
October 10, 2001, first show in USA after the 9/11 attacks, just one month after them.
Very emotional show, and a great performance.
But there's a lot of great shows in this tour.
Portland (the day that Joey Ramone died), any NY show after 9/11, then there's the Kansas show, which features one of only 3 full renditions of Peace on Earth plus the only time they played (or snippeted) When I Look at the World.
Vertigo Tour
Honolulu
Last show of the tour.
Very complete setlist, they invite some guy to play piano on Angel of Harlem but he only knows Wild Horses so they have to improvise the song on stage.
In the encore they're joined by Green Day to play Saints are Coming and by Pearl Jam to play Rockin in the Free World.
Any show when he hits the "l'amore" note right in Miss Sarajevo gets extra points for me.
360
Barcelona (july 02)
They're still playing lots of No Line songs, including the only time they played the album version of Crazy Tonight.
360 -part II-
Torino.
This tour changed a lot in a couple of years, so from the second half of the tour go with Torino, first show after Bono's surgery, and they debut 3 new songs (Stingray, Glastonbury and North Star).
There's an IEM but I prefer the audience recording.
There is a small chance I might be able to win tickets to Chicago through a contest!... Here's hoping... Anyway, welcome to the U2start community, Steve!
I'd like to put in my two cents on what the must-have shows are... There are a load of really great ones available so I'll try to limit myself.
Boy
1981-03-06 - Boston (the first show): really good sound, and a really strong performance of the material
October
1981-12-13 - Lido Beach: possibly the best-sounding October bootleg, very good show
Pre-War
1982-12-06 - London: One of my all-time favourite boots. The show isn't complete (the first two and final two songs are cut off) but what is there is in very good quality. This is really more of a showcase for the songs from October rather than for the few new War songs in the set. It's telling that the deluxe rerelease of October included a few songs from this performance. Some of the very best performances of Surrender, I Threw a Brick/Day Without Me, An Cat Dubh/Into the Heart, Gloria, Fire, I Fall Down, and A Celebration.
War
1983-02-28 - Edinburgh: I'll admit that War is probably my least-listened tour (not sure why...) so there are likely better shows out there but this is a really strong performance of the new material, with, again, the October material really shining.
The Unforgettable Fire
1984-10-23 - Nantes: Notable for including one of only a handful of Indian Summer Sky performances. A top-notch show where the War songs are great and a lot of the UF songs have a punkish edge to them.
The Joshua Tree
1987-04-29 - Rosemont: A real classic in the bootleg archives. Includes a performance of Ballad of Springhill and all the JT stuff is top-notch.
1987-07-04 - Paris: I'll admit I haven't listened to the bootleg on here, only the iTunes live release, but it's a terrific show. My personal favourite performance of With or Without You, while I Still Haven't Found, Exit, Trip Through Your Wires, Unforgettable Fire, Electric Co., and In God's Country are also all really really excellent.
Lovetown
1989-12-26 - Dublin: Of all the Point Depot shows, this one is probably the best. It's a truly incredible showing for Rattle and Hum and Joshua Tree songs.
ZooTV
1992-06-11 - Stockholm: My favourite bootleg of them all. The opening run of Achtung Baby songs is incredible. My favourite performances of Bullet the Blue Sky, Mysterious Ways, while One, Who's Gonna Ride, Zoo Station, Even Better, The Fly, and Stand By Me are also all standouts.
1992-06-13 - Kiel: The next show and my next-favourite bootleg. Probably the greatest performance of Love Is Blindness ever. Zoo Station, The Fly, and Until the End of the World are also extremely good. Rare snippet of Walk To the Water during Bad.
ZooTV: Zooropa
1993-08-28 - Dublin: My first bootleg! It's a great show, lots of fun. Mr MacPhisto is always wonderful. Tryin' To Throw Your Arms Around the World at its best, Love Is Blindness is also amazingly good.
PopMart
1997-05-06 - Eugene: Maybe not the greatest show on this list, but it's pretty overlooked. It's good to have because Do You Feel Loved and If God Will Send His Angels are still in the set.
1998-02-11 - Santiago: Awesome show. The real highlight is the performance of Mothers of the Disappeared, featuring ACTUAL mothers of the disappeared.
Elevation
2001-10-10 - South Bend: Really powerful show. Lots of ATYCLB tracks (Beautiful Day, Stuck, Walk On) are done very, very well. Bad with the snippet of Molly Malone is another show-stopper.
2001-11-09 - Salt Lake City: Another strong show. Some unusual tracks (Wild Honey, Please) in this one, and when they bring a fan onstage to play, he does a really good job! Elevation, Stuck, and Kite shine.
Vertigo
2005-10-11 - New York: A really strong show with some uncommon songs.
2006-11-29 - Saitama: The only one of the three Saitama shows I've listened to yet. It's very good. City of Blinding Lights, Beautiful Day, and One Tree Hill are probably the biggest highlights.
U2360
2011-07-09 - Montreal: Incredible sound, incredible show. The Fly, Stuck/space talk/Beautiful Day/Space Oddity, Zooropa, and One with a beautiful snippet of Hallelujah really make this show unmissable.
To me these are probably the four most essential bootlegs. After these, I recommend going on a voyage of discovery in the catalogue! 
Hope you have a good time! 