I will explain a result of Levelt and Turrittin that classifies linear systems of differential equations with coefficients in the field of formal Laurent power series (up to linear change of coordinates).
The conclusion is that every such equation admits a unique explicit "canonical form" after taking a root of the coordinate (i.e. passing to a ramified cover). There is an algorithmic procedure to obtain such canonical form, which depends on a finite number of coefficients.
This gives a description of linear systems of ODEs with meromorphic coefficients around a singularity (beyond just knowing the monodromy).