Linear finite-dimensional, continuous-time singular control systems with constant coefficients are considered. It is generally assumed that the controls are unconstrained and delayed. Using algebraic methods and the theory of singular linear differential equations, several necessary and sufficient conditions for controllability are formulated and proved. In the proofs of the main results a well-known canonical decomposition of singular control systems is extensively used. Moreover, these controllability conditions require verification of the rank of controllability matrices and have an algebraic nature. The paper contains also several remarks and comments on controllability problems for linear singular control systems and relations to controllability results in the literature. Finally, it should be mentioned that the results given in the paper are generalizations, for delayed singular control systems, of controllability conditions given in the monograph [J. Klamka
, Controllability of dyamical systems, Kluwer, Dordrecht (1991; Zbl 0732.93008
)] for regular delayed control systems.