High jump is certainly the athletics discipline that has gone through the most style changes: scissors, eastern cut-off, western roll, straddle, you name it. (An interesting article on the evolution of high jump techniques can be found at pinoyathletics).
Below is a photo of the greek champion (gold and bronze medal winner at the 1912 Olympics at standing long jump and high jump respectively, silver at both disciplines in the 1908 Olympics), Constantinos Tsiklitiras.
But the one style that came to dominate completely the discipline is what is called today the Fosbury flop. It is named after the 1968 Olympics gold medalist Dick Fosbury.
(In fact, Fosbury himself was calling his style "back layout" but the "flop" was more catchy and thus it caught on). I do like styles and techniques named after great athletes and naming the flop after Fosbury is unarguably well-deserved. (I always regret the disappearance of the "O'Brien" and "Baryshnikov" terms in shot put in favour of the more technical "glide" and "spin", although in the case of Baryshnikov, one can could argue that "spin" is much easier to pronounce).
Did Fosbury invent the style that has his name? The answer is an unambiguous "yes". After all no coach in their right mind would take the risk of such an unorthodox technique: it is always the athletes that come up with crazy ideas. What is clear in the case of Fosbury is that the time was right for such a revolutionary technique. The old sandpits for high jump has disappeared in favour of sawdust pits and they in turn were being phased our by foam rubber ones. If you jump in a sandpit your only choice is to land on your feet, as these photos from a present-day competition in Kenya can attest.
(You can find a link to the impressive video at anorak's site which has a superb collection of photos of high jumpers over the past century). With sawdust pits you can start taking risks. In this video you can see Fosbury jumping in such a wood-chip pit without hurting himself. However Fosbury was not the first to jump in the back layout style. A grainy photo from a high-school 1963 competition in Montana shows a young guy named Bruce Quande jumping in the flop style.
However Quande did not pursue high jumping after high school and thus his dabble with a new style was forgotten till the photo resurfaced, almost 50 years later.
The one athlete who independently invented the Fosbury style was the Canadian Debbie Brill.
There is a video of her in 1966, at the age of 13!, where one can see her jumping with the technique she had invented and which is known as the Brill bend. She went on to jump an indoor world record of 1.99 m with her technique and won the 1979 World Cup (which was the IAAF attempt at a world championship level competition before bona fide World Championships were introduced in 1983). She could have won a medal at the 1980 Olympics had Canada not declared boycott on the Moscow games.
So, who did invent the Fosbury flop? By 1963, when Quande was jumping in his style, Fosbury had already began experimenting on his own adapting the scissors style. Brill lived in the countryside and had no knowledge whatsoever of what was happening in athletics: she just jumped as was natural to her. So, both Fosbury and Brill should be credited with the invention and the "flop" is also a "bend".