As a certified shatnez checker myself who has learned the halachos I will tell you it is not secret.
I have certification from Lakewood- R’ Sayagh and Shochet. There are two “schools” essentially, meaning two ways shatnez checkers are trained and work. Lakewood teaches how to check, Williamsburg teaches how to take samples to send back to Williamsburg for checking. Regardless, the halacha is clearly gepaskened in shulchan aruch. If you have a garment with wool then you are half way to an issue. If you have a garment of linen, you are half way to an issur. The halacha requires one check a begged if it has one or the other to see if it has both. It’s that simple.
When I check a men’s suit I look at:
Button threads
Button holes
Reinforcements on the underside of all pockets
The dart
The Collar
The Canvas
The Shoulder pads
The inside of the waistband
Inside of the belt loops
Skirts and sweaters are much easier. You have less places it can be hiding. You look for selvedges, reinforcements and the end of each place it’s sowed together and decorations.
And of course you have to look at the fabric itself. Sometimes its not labeled accurately. Linen has a sheen look to it. There are other details used to discern the identity of the fabric or thread.
The big question is whether or not you can rely on a chazaka. Lakewood says no, Williamsburg says yes.
Rav Chaim was clear that you could, I also personally asked Rav Nota Greenblatt and he said “why not?”
The other issue is China- they will sow anything in as a reinforcement behind the dart or pockets. Meaning, there can be unintentional shatnez. Otherwise, expensive suits use linen in the collar and canvases because it provides a superior drape.