Abstract <p>We consider linear systems of ordinary differential equations of arbitrary order with rectangular coefficient matrices. It is assumed that at any point in the domain the matrix that multiplies the highest derivative of the desired vector function has incomplete rank. We study solvability issues and search for the structure of general solutions to such systems. Particular attention is paid to systems that have singular points in the domain. We assume that a point is singular if the system under study has no solutions on the segment containing this point, or the solution to the corresponding initial value or boundary value problems is not unique, or if we can observe the change in the solution manifold dimension. We attempt to formalize the concept of a singular point and thereby provide their classification. We discuss application of the least squares method as the most suitable technique for numerical treatment of such problems. The residual functional is chosen in Sobolev spaces determined by the properties of the internal structure of the system. All theoretical results are illustrated by examples.</p>