  The representations of a $k$-graph $C^*$-algebra $C^*(\Lambda)$ which arise
from $\Lambda$-semibranching function systems are closely linked to the
dynamics of the $k$-graph $\Lambda$. In this paper, we undertake a systematic
analysis of the question of irreducibility for these representations. We
provide a variety of necessary and sufficient conditions for irreducibility, as
well as a number of examples indicating the optimality of our results. We also
explore the relationship between irreducible $\Lambda$-semibranching
representations and purely atomic representations of $C^*(\Lambda)$. Throughout
the paper, we work in the setting of row-finite source-free $k$-graphs; this
paper constitutes the first analysis of $\Lambda$-semibranching representations
at this level of generality.
