What is commonly called "random mutation" does not in fact occur in a mathematically random pattern. The process of genetic mutation is extremely complex, with multiple pathways, involving more than one system. Current research suggests most spontaneous mutations occur as errors in the repair process for damaged DNA. Neither the damage nor the errors in repair have been shown to be random in where they occur, how they occur, or when they occur. Rather, the idea that mutations are random is simply a widely held assumption by non-specialists and even many teachers of biology. There is no direct evidence for it.
On the contrary, there's much evidence that genetic mutation vary in patterns. For instance it is pretty much accepted that mutation rates increase or decrease as stress on the cells increases or decreases. These variable rates of mutation include mutations induced by stress from an organism's predators and competition, and as well as increased mutations brought on by environmental and epigenetic factors. Mutations have also been shown to have a higher chance of occurring near a place in DNA where mutations have already occurred, creating mutation hotspot clusters—a non-random pattern.
While we can't say mutations are random, we can say there is a large chaotic component, just as there is in the throw of a loaded dice. But loaded dice should not be confused with randomness because over the long run—which is the time frame of evolution—the weighted bias will have noticeable consequences. So to be clear: the evidence shows that chance plays a primary role in mutations, and there would be no natural selection without chance. But it is not random chance. It is loaded chance, with multiple constraints, multi-point biases, numerous clustering effects, and skewed distributions.
So why does the idea of random mutations persist? The assumption of "random mutation" was a philosophical necessity to combat the erroneous earlier idea of inherited acquired traits, or what is commonly called Lamarckian evolution. As a rough first-order approximation, random mutation works pretty well as an intellectual and experimental framework. But the lack of direct evidence for actual random mutations has now reached a stage where the idea needs to be retired.
There are several related reasons why this unsubstantiated idea continues to be repeated without evidence. The first is fear that non-random mutations would be misunderstood and twisted by creationists to wrongly deny the reality and importance of evolution by natural selection. The second is that if mutations are not random and have some pattern, than that pattern creates a micro-direction in evolution. And since biological evolution is nothing but micro actions accumulating into macro actions, these micro-patterns leave open the possibility of macro directions in evolution. That raises all kinds of red flags. If there are evolutionary macro-directions, where do they originate? And what are the directions? To date, there is little consensus about evidence for macro-directions in evolution beyond an increase in complexity, but the very notion of evolution with any direction is so contrary to current dogma in modern evolution theory that it continues to embrace the assumption of randomness.
By retiring the notion of fully random mutations we can gain some practical advantages. The idea that mutations have a bias can be exploited to more easily to engineer genetic processes using those biases. We can better understand the origin of disease mutations, and to remedy them. And with this new understanding we can better resolve some of the remaining mysteries of macro evolution. An important part of retiring the idea of random mutations is to realize that the chance element operating in mutations is not "imperfect" randomness, but rather contains a bit of order that is generative—a small something that can be used by either us or natural selection. What it is used for, or can be used for, is wide open, but we'll never get there if we cling to the idea that mutations are random.