This is a fun one. We’re always told that mutations occur randomly throughout the genome, but this idea kind of misses the point that they are, in their own way, “predictably random.” At different parts of the genome, mutations occur at different frequencies. On the individual level, different types of mutational causes are more frequent than others. Each site along the DNA has a mutational variance based upon conditions the region is subject to. This is why the various “molecular clocks” need to be calibrated separately from one another. We have many molecular clocks, indeed, we are limited only by the number of common genes between organisms. Indeed, the LUCA (Last Universal Common Ancestor) possessed at least:
rRNA and tRNA genes
DNA-binding genes for topoisomerases, ligases, and repair proteins
glucose metabolism genes (hexokinases, at least)
80 genes, all told. This is useful if you’re comparing a specific bacteria to a specific bird, but many species share more genes than these. Mammals, for example, share more genes with each other than they share with yeast and plants. This is only to be expected, but I’m getting off topic. Different genes mutate at different rates, and these rates vary based upon several factors. One of which being the DNA repair and replication genes of that organism. Some repair and replication proteins are notably more efficient at repairing or replicating DNA with higher fidelity. This variance in fidelity plays a role in mutation rates. Broadly speaking, however, the introns of these genes vary little within families and orders due to the selective pressure to have functional DNA repair and replication genes. The rates are about 1/50th that of silent stretches of the genome.
Another point is that mutation rates are not exactly correlated to years, but to generations, so a shortening in generation time will lead to an increase in the mutation rate and vice versa. Yet, once again, the increase will be more pronounced in the silent stretches of DNA than in the functional stretches.
Additional compounding factors include the methylation/acetylation states of histones (at least in eukaryotes) which allow more or less exposure to the repair proteins found within the cell. For this reason, (generally speaking) genes bound by highly methylated histones tend to undergo higher mutation rates than those around demethylated (or specifically methylated) histones.
Moreover, the different molecules which comprise DNA differ in their interactions with neighboring/paired molecules. Two thymines next to one another can fuse to form dimers if exposed to UV light. Cytosine can spontaneously deaminate to form uracil. Mismatched pairs vary in frequency due to variations in partial affinity.
These are certainly not ALL of the problems with the idea of absolutely random mutations, but make for a decent start on the issue. We have to understand that while specific mutations occur randomly in terms of if or when they will happen, the rates of mutation differ due to various compounding factors.