How do people check the type of defect in a crystal?
As you said there are many different kind of defects in crystals. There are basically two different methods to find the Crystallographic defect.


Mathematical classification methods
Computer simulation methods


Mathematical classification methods:

A successful mathematical classification method for physical lattice defects, which works not only with the theory of dislocations and other defects in crystals but also, e.g., for disclinations in liquid crystals and for excitations in superfluid $\ce{^{3}He}$, is the topological homotopy theory.

Computer simulation methods:

Density-functional theory, classical molecular dynamics and kinetic Monte Carlo simulations are widely used to study the properties of defects in solids with computer simulations. Simulating jamming of hard spheres of different sizes and/or in containers with non-commeasurable sizes using the Lubachevsky-Stillinger algorithm can be an effective technique for demonstrating some types of crystallographic defects.



Source: Wikipedia