Psychobilly was a bizarre mutant hybrid of punk and rockabilly which emerged at roughly the same time as goth. It's worth mentioning here because there was a certain amount of crossover between the two emergent subcultures- some bands such as Alien Sex Fiend, Sunglasses After Dark and the Cramps appealed equally to goths and psychobillies, and some of the "punkier" goths had psychobilly leanings.
For further background, here's the article on Psychobilly from Ted Polhemus' "Streetstyle":
At first glance it is hard to imagine a more unlikely combination than Punk and Rockabilly, but the Psychobillies made a virtue of such apparent incompatibility. At the wonderfully named 'Klub Foot', the West London venue where the Psychobillies first came together as a subculture, their fusion of 1950s Americana and 1970s British Punk seemed both obvious and inevitable.
To make the connection one must forget the soft drizzle of sentimentality which in the end became all too typical of the Rockabillies (Elvis singing about Teddy Bears in Vegas) and go back to the angry, licentious snarl of their early days. From this perspective it is clear that the thumping beat, the in-your-face sexuality, the deliberate shunning of prissy sophistication and the greasy quiffs of the early Rockabillies were in tune with Punk's gutsy spirit of raw rebellion. The Punks simply added a stylistic extremism, an assumption of gender equality and a fetishistic trashiness which could not conceivably have existed in Memphis in the mid-fifties. The common denominator is rock n' roll energy in its purest form.
Although the slezoid music and style of the American post-Punk band The Cramps was clearly an inspiration, the first 100percent-proof Psychobilly band was The Meteors, which formed in South London in 1980. With musicians consisting of one Rockabilly, one Punk and one psychedelic horror enthusiast, The Meteors constituted a complete microcosm of the subculture which would almost immediately form around it.
By 1982, with the opening of Klub Foot, the Psychobillies were more than simply the followers of a cult band. Their style has bee, termed 'Mutant Rockabilly' and it is an apt description - with cartoon quiffs sometimes dyed green or purple and always thrust out far beyond the expectations of gravity, aggressive studded belts and Doc Martens, shredded, bleached jeans and leather jackets painted with post-nuclear-holocaust imagery. Here were creatures straight out of tacky comic books or ketchup-splattered horror movies brought to life (?) and waiting patiently for the last bus to Planet Zorch. Needless to say, such an extreme styletribe never reached an enormous size and its bands (in time including the likes of Guana Batz, Demented Are Go, Batmobile and the truly unbelievable King Kurt) never appeared on TV's Top of the Pops. It did, however, quickly acquire members throughout most of Europe (especially Germany, Italy and Spain) and a large, dedicated following in Japan.
Stylistically, the Psychobillies' principal effect seems to have been on the Rockabillies - causing a shift towards battered denim workwear and away from fancy suits and pristine footwear. From there (and it should be remembered that the Rockabilly movement was huge in Britain in the early eighties) this look moved into the streetsmart mainstream in the form of the 'Hard Times' look. At one level the Psychobillies exhibited an alarming fixation with violence and wanton destruction, but this was always tempered by a wonderful, surreal sense of humour, which made you smile, even as you crossed hurriedly to the other side of the street.
Note that psychobilly and goth arrived at much the same time- Klub Foot opened the same year as the Batcave.
 Apparently Ted Polhemus is wrong here- King Kurt appeared on TOTP with "Destination Zululand". Thanks to Rose Francis for that one.