We present the results of a simple but robust morphological classification of a statistically complete sample of 108 of the most X-ray-luminous clusters at 0.15 ≤z≤ 0.7 observed with Chandra. Our aims are to (a) identify the most disturbed massive clusters to be used as gravitational lenses for studies of the distant Universe and as probes of particle acceleration mechanisms resulting in non-thermal radio emission; (b) find cluster mergers featuring subcluster trajectories that make them suitable for quantitative analyses of cluster collisions; and (c) constrain the evolution with redshift of the cluster merger fraction. Finally, (d) this paper represents the third public release of clusters from the MAssive Cluster Survey sample, adding 24 clusters to the 46 published previously. To classify clusters by the degree of relaxation, we use the projected offset of the brightest cluster galaxy from the peak (or the global centroid) of the X-ray emission as a measure of the segregation between the intracluster gas and dark matter, and also perform a visual assessment of the optical and X-ray morphology on all scales. Regarding (a), we identify 10 complex systems likely to have undergone multiple merger events in the recent past. Regarding (b), we identify 11 systems likely to be post-collision, binary, head-on mergers (BHOMs), as well as another six mergers that are possible BHOMs but probably harder to interpret because of non-negligible impact parameters and merger axes closer to our line of sight. Regarding (c), we find a highly significant increase with redshift in the fraction of morphologically disturbed clusters (and thus a clear decrease in the number of fully relaxed systems) starting at z∼ 0.4, in spite of a detection bias in our sample against very disturbed systems at high redshift. Since our morphological diagnostics are all based on imaging data and thus sensitive to projection effects, the measured merger fractions should be considered lower limits and our list of mergers incomplete, as we are likely to miss systems forming along axes close to our line of sight. A larger sample of clusters with high-quality X-ray data in particular at high redshift will be needed to trace the evolutionary history of cluster growth and relaxation closer to the primary epoch of cluster formation z∼ 1.