Little bit of a followup on the "Series Solutions of ODEs in Maple" online seminar.

According to Mathematical Methods for Physicists, 7th Edition by Arfken, Weber and Harris,

Pages 343-345,

Singular points are classified as regular or irregular

Irregular points are called essential singularies.

They show how to apply these to famous differential equations in Quantum Mechanics and other physical applications (examples given in Farlow's Partial Differential Equations for Scientists and Engineers).

In Section 12.1 of Mathematical Methods for Physicists, the complex series Laurent expansion (chapter 11 of the book) is applied to generalized to the complex plane (see Saff and Snider Fundamentals of Complex Analysis for Mathematics, Science and Engineering, 2nd Edition).  Not too sure how Maple handles contour integrals though.

It seems that a regular point is the same as a ordinary point, as per Elementary Differential Equations and Boundary Value Problems, 8th Edition by Boyce and DiPrima, Chapter 5.