Colouring Steiner systems with specified block colour patterns

We consider colourings of Steiner systems S(2,3,v) and S(2,4,v) in which blocks have prescribed colour patterns, as a refinement of the classical weak colourings. The main question studied is, given an integer k, does there exist a colouring of given type using exactly k colours? For several types of colourings, a complete answer to this question is obtained while for other types, partial results are presented. We also discuss the question of the existence of uncolourable systems.