BB spindle length standard for single speed bike?
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.