Kemper WD, Rosenau RC. 1986. Aggregate Stability and Size Distribution. In: Klute A, editor. Methods of soil analysis. Part 1. Physical and mineralogical methods. Madison, WI. p 425-42.
The most common is the wet sieving using wet sieving apparatus following the method of Kemper WD, Rosenau RC. 1986. Aggregate Stability and Size Distribution.
These two papers summarize the methods and provide some theoretical basis for the measurements:
LE BISSONNAIS, Y. (1996), Aggregate stability and assessment of soil crustability and erodibility: I. Theory and methodology. European Journal of Soil Science, 47: 425–437. doi: 10.1111/j.1365-2389.1996.tb01843.x
LE BISSONNAIS, Y. and ARROUAYS, D. (1997), Aggregate stability and assessment of soil crustability and erodibility: II. Application to humic loamy soils with various organic carbon contents. European Journal of Soil Science, 48: 39–48. doi: 10.1111/j.1365-2389.1997.tb00183.x
There are numerous ways to measure aggregate stability, depending on the disruptive forces applied, ...Some of them use machines, so increase the reproducibility...
I think you should take into account why do you want to measure the aggregate stability and what for. And depending on that, you could choose the method that better will answer your question.
The simplest way is the cell phone app called SLAKES. It takes 10 minutes and seems to be very accurate. I am yet to test it. Article Evaluation of SLAKES, a smartphone application for quantifyi...