Mild steel is just another name for ordinary carbon steel - it is iron alloyed with carbon, which adds strength. There are various grades, but "special" doesn't mean anything unless the "specialness" is specified. For example, garden-variety carbon steel pipe that you can buy at the hardware store is grade A53, and there is an ASTM standard for it. What prompts your question?
I too have ran into this. Usually it reads, “The stern frame is of special quality mild steel and is of hollow or dished section, constructed in two pieces, 63ft and 37ft 4in, in length respectively.” From what I gathered this 'special mild steel', is of Siemens-Martin.
“All these castings are of Siemens-Martin mild cast steel with the exception of the rudder stock, which is of forged ingot steel.” Marine Review Vol. 41, 1911
It had considerable advantages over the Bessemer technique. Bessemer's conversion took place in half an hour, and quality control was imperfect. The Siemens-Martin process took fourteen hours to complete, allowed extensive chemical analysis, and therefore permitted precise control of quality. Moreover, since it worked ion the principle of mixing solid malleable iron into a molten bath of pig iron to produce cast steel, it also permitted the addition of scrap, and hence the re-utilisation of old malleable iron rails and plates for conversion to steel. - The Development of the West of Scotland
"Siemens-Martin" is the just the old name for the open-hearth furnace process for making steel. The name "Siemens-Martin" faded away, and it just became know as open-hearth. Open hearths were used for many decades, but were retired in favor of the basic oxygen furnace. Open hearths emit a nasty amount of air pollution.