Choosing an easement curve 

I'd like to be clear about:
(a) which curves are actually used in practice ...
(b) in the various railways administrations; not just North American but also others, e.g. Australian, European, Indian etc.  (This would help with NPOV.)

My main concern is with (a).  Although the article mostly discusses Euler spirals, it seems to give the impression that's the only curve used, under varying names.  However, consider this quotation (from the AREMA reference in the article):
There are several different types of mathematical spirals available for use, including the clothoid, the cubic parabola and the lemniscate.  Of more common use on railways are the Searles, the Talbot and the AREMA 10-Chord spirals, which are empirical approximations of true spirals.
This reference is, of course, discussing common use on American railways; but even in this limited context, it appears there are at least three ideal mathematical spiral forms involved: "the clothoid, the cubic parabola and the lemniscate", along with various practical approximations to them.

The article would ideally give the history of the use and adaptation of these various spiral forms.  Doing so would provide a solid illustration of how technological innovation was both necessitated and encouraged by the Age of Rail - a valuable insight into the social impact of rail.

I'm guessing that the AREMA reference probably continues on from the quote above to give some further detail about those approximations, their inventors and their history; but it may not.  In any case, can anybody provide suitable references to help flesh out this story?