The phrase "Random Mutation" does not entirely project a clear concept.
There are many kinds of mutations caused by a variety of molecular/cellular mechanisms.
Location of mutations: The positions of mutations within a genome can appear random at some level of resolution.
I've been involved in a variety of mutageneses using a variety of methods (X-rays, UV, P-elements, ENU (vertebrate chemical mutagen), Psoralen (a chemical that can cross links DNA it they are together exposed to UV). They each have different advantages and limitations, including various ways of not making random mutations.
The most often ideal goal of a mutagenesis is (starting from a known isogenic (all individuals in the population are genetically identical)) to generate mutations at random throughout the genome. This distribution of new mutations that are generated through what ever skewed distribution in the genome. The rate new mutations are generated at, is controlled, such that number of mutations in a single germline cell, doesn't generate more mutations, per germline cell, can be identified through genetic breeding trips and screening/selections.
The ideal desired use of a good mutatgeneises is to do a saturation mutatgenesis. In a saturation mutagenesis, you collect so many mutations that by reasonable chance, you have collect mutations in every gene that your screen (looking for mutants) can capture phenotypes of. The mutations that are found, define pathways by the genes, identified by mutations, that the mutagenesis finds.
This provides me with a complex view of random mutations.
Realize that a real world (or natural) rate of spontaneous mutation will be a summation of many different processes going on, in the uncontrolled "normal" or "real" world.
Overall Rate:
In a lab, things can be separated and controlled. But that doesn't happen in the "wild". In some places, at some times, different mutagenic rates will vary for the different mutation generating mechanisms.
In the sense that its an uncontrolled and unanalyzed mess, the resulting total rates can be considered, first look random.
Chemical and radiation mutagenic exposures can easily be controlled in the lab.
In nature, they could vary widely from place to place, and from time to time.
Location is not always random:
Some mutational processes target or avoid certain genetic sites. There are lots of plausible proposed reasons why, but I don't know of clear demonstrations of most of them.
Biological Mutagenesis, like the transposition of
P-elements within a genome, will often favor particular genomic sites (defined by the sequence they insert into). This is presumably due to particular molecular cellular reasons limiting or block access to particular genomic sites. This skewing of mutation by this particular method is often quite local, which might leave other regions of a particular gene available for attack. P-element use is well controlled by researchers to mutate and ID genes. Although, it might not produce a pure random collection of mutations, it provides a lot and the mutations are easily tied to the sequence they are affecting.
Chemical and radiation caused mutations are considered to have a more random distribution of the sequences they affect (nothing should get in the way of X-rays, many mutagenic molecules are not large (like a protein) and therefore should not be so easily blocked).
(However, they do not have a useful marker linked to the mutated sequence, so they require more work to link the genetic mutation (determined by genetic patterns of inheritance) to a particular molecular sequence in some region of the genome (which is an important component of molecular analysis).)
Types of Mutations Produced:
Different causes of mutations make different kinds of mutations. There is a specific distribution to the innds of mutations each method will produce: point mutations, break point mutations, short deletions, sequence insertions, a mixture of each.
So the amount, distribution through the genome, and the kinds of mutations could vary uncontrollably in a non-lab situation.
It might look random mutations, but there would probably underlying non-randomnesses in its distribution.