The Cramps - NY/LA band
The Birthday Party - Melbourne/London/West Berlin band
X - LA band

BP for me with no contest

The Birthday Party by a mile. Granted, it took the band a sec to shake off some of the more lightweight new wave/postpunk moves from their Boys Next Door period. But shit, once things got rolling... Aggressive, stylish, and downright rocking for such a sour-faced Anglo (ok, Australians based in Britain) band from that period. Cave, Howard, and Pew were all pretty singular players. Prayers on Fire, Junkyard, and those last two EPs are a hell of a run, made even better by the live album that came out in the '90s.

The Cramps were good fun, especially up to and including the Kid Congo era, but then they kinda calcified. Still, a very important band, and Lux and Ivy (plus Bryan Gregory, to a lesser extent) were as distinctive as anyone in the Birthday Party. Real innovators as far as trash-culture retro-garage punk goes—although unless you count stuff like Pussy Galore or the Black Snakes, I don't love a lot of the other music that's cut from that cloth. (And come to think of it, both of those bands sound equally indebted to the Birthday Party!) The Cramps are kind of rollicking by comparison.

The best moments in X are all about Billy Zoom and, to a lesser extent, DJ Bonebrake. There are certainly good things on the first two records, but you also get the impression that this band listened to its management and/or record label. At least, a hell of a lot more than unruly fucks like the Birthday Party or the Cramps did. And thus, there is also a lot of slick, awkward, and downright no-good in their catalog. Not useless, just very uneven.

I love all 3, but it's The Cramps.

BP....so much of their discography still sounds super innovative and unlike anything before or since. Obviously there's been bands who've picked up on their style and incorporated other elements (Oxbow would be a good example) but their is visceral intensity and WTF factor that still feels unmatched.

As for Nick Cave's post BP career, my personal preferences are:

Music after his son died > Grinderman > 80s and early 90s Bad Seeds> attempts to become Leonard Cohen.

BP has something edgy and ground breaking. I'm sure in it's time and place it was a magnificent mind fuck. For me looking back over a stack of records and decades of rock history they don't have the memorable songs. An exciting racket but within each album a lot of interchangable material that just kind of blares without leaving a specific impression. (Major waffles for the fact that I've listened to them the least).

Cramps- almost the opposite. Way less of a blaring smear of energy and instead a spacious surf vibe that oozes cool and low key menace. They're like a fucking comic book. Their best stuff is like a lot of that old punk, or even old R&B 45s where the title is the hook, is the theme, is the gimmick. Maybe not a lot of emotional depth but at best a kind of crystalized rock and roll perfection in all its campy glory.

X- my fav of the bunch. The vocal meld of those two has no competition in punk history. They're like The Everly Brothers of the apocalypse. If anything in post 60s rockabilly was going to be cool they sucked it all up. Los Angeles and Under the Big Black Sun hold more durable tunes than Birthday Party or Cramps whole discography even if they were more conventional than the others.

Tend to suspect that only one of these bands have this one in it's back pocket.

Cramps are nice, but a bit of a novelty act to me.
Birthday Party is very cool, but specialize in that very discordant style.
X is probably my favorite Punk band, terrific songs and playing, and totally without pretense. I like that era where you would have some hotshot rockabilly players adding licks in Punk/New Wave bands (East Bay Ray, Billy Zoom, James Honeyman-Scott, Elliot Easton....I'm sure there are more). That never really came around again.

X

The Cramps, no contest. They might have been the perfect rock band. Easily some of the best shows I have ever seen, I think I saw them three times.

Agree about BP having a time and place, they did something every different, and very well, but I don't often want to listen to them.

X was my first concert. Great band, great songs, great lyrics, intimate and vulnerable. Doe and Cervenka are to punk what Buckingham and Nicks were to Adult MOR. NOt a fan of much X after Under the Big Black Sun. Have seen both Doe and Cervenka solo, and the Knitters, and the Original Sinners. Still have some love for them.