Theoretical arguments are presented which suggest that each advance of Muller's ratchet in a haploid asexual population causes the fixation of a deleterious mutation at a single locus. A similar process operates in a diploid, fully asexual population under a wide range of parameter values, with respect to fixation within one of the two haploid genomes. Fixations of deleterious mutations in asexual species can thus be greatly accelerated in comparison with a freely recombining genome, if the ratchet is operating. In a diploid with segregation of a single chromosome, but no crossing over within the chromosome, the advance of the ratchet can be decoupled from fixation if mutations are sufficiently close to recessivity. A new analytical approximation for the rate of advance of the ratchet is proposed. Simulation results are presented that validate the assertions about fixation. The simulations show that none of the analytical approximations for the rate of advance of the ratchet are satisfactory when population size is large. The relevance of these results for evolutionary processes such as Y chromosome degeneration is discussed.