Another in the long line of cycling myths goes approximately like this: "When buying shoes for cycling make sure you get the ones with the stiffest sole you can find." The argument seems to be that the energy is lost by the flexion of the sole, so the one that flexes the least is the best. I must admit that this is a very convincing myth. I myself had fallen for it until very recently.
But firstly, a clarification of the terms. If by "cycling shoes" we mean special shoes that are fixed to a pedal (SPD or similar clipless system) than this is not a myth but a reality. In this case the force on the pedal is transmitted through small area and if the sole were not strong (=stiff) it would quickly be ruined. However, I think that the stiff-sole-myth had been around before the clipless system hype, so it doesn't descend directly from it (btw, for the reasons explained above the clipless system is not an option for me).
A recent experience convinced me that stiff sole used with ordinary pedals (no clips or straps) might be unfavorable. One Saturday I rode 150 km with two friends in a relatively fast pace. During and after the ride I felt great, the bike fit was perfect, I had no pains whatever. I felt as if I was at the peak of fitness. Next Sunday I rode 80 km "recovery" ride, same bike, same clothes, different shoes. To my surprise on that ride nothing seemed right: I had back pains and the saddle felt uncomfortable. On Saturday I had trainers with soft rubber sole, on Sunday I had shoes for in-door football (which, btw, were up to now my touring shoes) with stiff sole which became sleek and slippery during three years of use. So I am quite convinced that my uneasiness on Sunday was caused by the constant corrections of the position of my feet because the soles of the shoes were slipping on the pedals. If the soles were a bit more flexible they would grip the pedal preventing the slip.
My conclusion is therefore that the non-slipping contact between the pedal and the sole is much more important then the stiffness of the sole; and this is better attained if the sole is somewhat softer.