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Mutations arise from the continual random damage that the genes of a living creature are known to be subject to. It's generally agreed that when there's damage to something as complicated as that--the blueprint for the development of a living creature--many more of these mutations to genes will be harmful than beneficial. Harmful and beneficial mutations in the proportions as they arise would quickly lead to extinction. Of course harmful mutations make living creatures less likely to survive to reproduce so they're less likely to be inherited. But, according to the Modern Synthesis, even getting rid of all the harmful mutations would not by itself drive evolution. For that, you must have Beneficial Mutations. But do they exist?
Identifying Beneficial Mutations is not easy. Suppose you observe a gene being mutated at random, how can you tell whether or not that mutation is Beneficial? Really, you can't. Only millions of years later can you tell whether or not it has contributed to the evolution of a new species. Take a mutation making an elephant more salt tolerant, is that a Beneficial Mutation? That depends on whether, millions of years from now, elephants evolve into marine creatures. By that time, it'll be hard to know if the genes responsible for salt tolerance originated through random mutation or some other cause.
Mutations can be induced in the laboratory that confer a benefit on a species of living creatures, but that is not the same as accounting for how species evolve into one another in the wild. What you need for that is a sighting of a Beneficial Mutation in the wild, but that has so far proved elusive. Of course, the problem can be very simply solved by simply declaring that all evolution is due to Beneficial Mutations. Then the fact that evolution happens means Beneficial Mutations must exist and you've proved your point. But that's not the same as an actual sighting.
Perhaps we can draw lessons from the fate of other creatures supposed to exist or not to exist. The unicorn was for centuries confidently declared to exist but persistent failures to sight one led to them finally being declared to be mythical. Black swans, on the other hand, at first assumed to be mythical, were subsequently found, in Australia. The lesson seems to be, either you find a creature and it's not mythical, or if you can't find it it's mythical. By that standard Beneficial Mutations qualify as mythical.
Actually, sketches of the Beneficial Mutation do occur, in equations penned by the mathematician Ronald Singer in the 1930s. They manifest as entities summoned up, as if by the dark arts, as statistical possibilities. Logically, Singer said, random damage to a blueprint, no matter how simple or complex the blueprint is, could conceivably result in the blueprint being improved. The incidence might be vanishingly small but it is not, mathematically, zero. By inserting into his equations a symbol for this vanishingly small possibility, Singer brought into existence the Beneficial Mutation, which he then had natural selection so favor over harmful mutations that over millions of years this could become a mechanism of evolution. Thus was population statistics born, and the Modern Synthesis launched.
Fortunately, the issue is of very little significance. Since everyone is satisfied that evolution has been accounted for through population statistics, whether or not the Beneficial Mutation actually exists doesn't matter. In fact, having someone declare it had become mythical would be very disturbing, to no good purpose. Anyway, even if Beneficial Mutations did exist, and continued to be selected for generation by generation, each for some specific small benefit of its own, there's no assurance they would over time spontaneously come to code for some other benefit, such as participating in the machinery needed to develop some complex new feature, such as feet turning into flippers.
The point is this: as science matures it's important we all pull together to support the existing structure, or it could all collapse. If that means turning a blind eye to whether some supposed entity does or does not exist, so what!