After a short mathematical introduction to repeated quantum measurement processes, we will focus on the probability distribution of the sequence of measurement outcomes. We will limit ourselves to the case where all measurements take values in a finite set. As we shall see through a series of examples, the resulting distributions range from familiar (i.i.d., Markov, ...) measures to highly singular, non-Gibbsian ones. We will explore some of the singularities from the point of view of the large deviations of entropy production.