BB spindle length standard for single speed bike? Is there a common spindle length for typical fixie/single speed bikes? Namely a bike with:

a 68 mm BSA bottom bracket shell
a square taper bottom bracket cartridge bearing
120 mm dropout distance
single chainring and single freewheel/fixed cog (flip-flop)
a wide single speed chain
fairly recent (2010+) cheap steel frame bike (sold expensively for being hip)

This question would be hopeless for bikes with doubles and derraileurs. However the considerable limitations on fixie chain lines might also have lead to a de-facto standard or a common spindle length most bikes share. The latter in particular, as these bikes appear to be standard mass-products that merely differ by colour, brand, and price tag.
There is no 1 standard square tapered bottom bracket axle length for track frames.
The crankset dictates the bb axle length, it's not as varied as with crankset with gears, but there is more than one.
The range is much smaller than with gears, track bb axle lengths mostly go from 103mm - 112mm (there are smaller and longer for less common cranksets).
I think 103mm and 107mm are the most common sizes, given that their respective cranksets would be the most common cranksets. Although, this doesn't mean that there aren't plenty of others.
Examples:

Factory5 Pista Lattice Chainset - 103mm
Stronglight Track 2000 Chainset - 107mm
Sugino SG75 Bottom Bracket - 109mm
Campagnolo Record Pista Bottom Bracket  - 111mm

There are many more examples, but I verified these by using the Velodrome.shop website.