Well, the Wiki entries (Rail Track and Rail Profiles) is fairly lame, so here's a few interesting links in regard to strap rail:
Colonel Melnikov's Report (~1840)
Gives some interesting info on strap rail usage of US railroads at the time (in addition to T rail), including dimensions & weaknesses
A discussion of Southern Michigan straprail trackage with an (all too small) cross section, and also a image & description of the dreaded 'Snake-Head' syndrom (i.e. when the iron strap worked loose, got snagged by the wheels of the car, and then was pushed up and through the car floor.
Image of strap rail
Googling around, you find general indications that strap rail for 'steam' rail service in the US disappeared as soon as conditions for T rail replacement existed (1840s - 1850s depending on the road and its finances), but in certain light-duty instances (such as Michigan streetcars or Western mines, strap rail remained in use well into the 1890s (if not later, although I couldn't find any info on this).
At any rate, strap rail keep speeds low, loads light, and passengers in terror of snake heads, so it went as soon as the road could afford to (and was able to) replace it.