My argument against the present evolution theory is that it misses some important evolution mechanisms and cannot explain the origin of many new species.
Evolution, that it, species develop from earlier species, can be seen in the fossil record and it is not in doubt. It is also not in doubt that random mutations and geographic separation do eventually lead to new species by splitting the original population and allowing the separated populations to drift away from each others genetically. The drift may be driven by selection (like in natural selection and sexual selection) or by random drift. Given enough time separated populations cannot anymore reproduce and form separate species.
Random mutations can create major changes in the expression of genes. This can happen e.g. if a gene turns on or off other genes. Thus, it is possible to breed a population of animals having no tails though the original population had tails. But the genes for building a tail are still in the genome, only turned off, so even a single point mutation may be enough to turn the genes on and the population again grows tails.
There are other forms of mutations than point mutations affecting only one DNA-base, such as insertions, duplications and deletions of parts of the genome. Mostly these mutations take or remove parts of the genome from the population pool and cannot create totally new gene portions. But there is also the possibility of recombination of DNA from different species. As the DNA of a different species must penetrate a cell, this mechanism mainly happens with viruses: two virus species that infect the same organism may sometimes recombine into a new virus species. It is also possible that a DNA-virus recombines with the DNA of the host and this can move genes from the virus to the host. With humans at least the human papilloma virus can recombine with human DNA. There is some indication that genes may have been moved in this way between humans and animals living in close proximity to humans: the human DNA has more shared genes with cows, cats and mice than with such ancestors of these animals that do not live close to humans.
Additionally there are crosses between species caused by sexual reproduction. Humans seem to have genes at least from Neanderthal and Denisova men, and from some more archaic humans.
Yet, these known mechanisms are almost certainly not the only ones as they cannot explain the fast evolution phases of new species in the fossil record after major extinction events. Especially one can look at the evolution of mammals. All mammals have a special gene for producing milk. It codes a protein that did not exist in other animals that are not mammals, but one can find the likely ancestral gene. The ancestral gene differs quite much from all milk genes in different mammals (the genes are very similar) and a question is how the milk gene developed. Apparently it developed in a relatively short time. The original gene did not produce milk. It had another job. Assuming that the original gene mutated to the milk gene through random mutations we get to the problem that no selection process could guide these mutations. After a few mutations the original gene does not any longer correctly perform the original task it had. As this task has not disappeared, some other gene must do the original task. Usually what happens is that first the original gene gets duplicated by a duplication mutation, then one of the copies performs the original task while the other is unnecessary and can mutate by random mutations. The copy that mutates does not perform any useful task, as it is too far from the original form to perform the original task and has not yet mutated so far that it could do some other useful task. We can deduce that the intermediate mutated forms do not perform any useful task because we do not see these intermediate terms in any animals doing some useful job. Only few DNA-code portions can do anything useful. Starting from a useful gene we need to make quite many point mutations before ending up with another gene that is useful for anything. Because the intermediate mutated genes do not do anything useful, they are harmful and they become turned off. That is, the mutating gene becomes a pseudogene.
When the mutating gene is a pseudogene, it is not active and no selection process, whether natural selection, sexual selection, or whatever selection, can act on it. The mutation is purely random and we can calculate the probability of the gene mutating a number of steps from the original gene directly from the binomial distribution. All that is needed for calculating the probability of reaching any useful mutation is to know the mutation rate of point mutations, the length of the mutating portion (i.e., the length of the protein coding part of the gene), the number of mutations needed (in the milk gene example, the distance between the original gene and a generic milk gene), the size of the population and the time available. I have done this calculation and the result is that it is extremely improbable for the protein coding part of the milk gene to have developed from its ancestor in this manner.
Thus, there has to be another manner. I have earlier suggested that maybe the mutation rate was much higher during this time and I will elaborate this hypothesis. Basically, there is not much that can be changed. For sure the mil gene has developed from its ancestor by mutations. There is no other way. Insertions and deletions do not explain the differences between these genes. There have been many point mutations. The population size cannot be increased enough to make such a chain of mutations probable. What remains is that either there is some guiding principle acting on non-active genes, which seems impossible, or the mutation rate has been temporarily much higher. Therefore the latter assumption remains as the most likely one.
The idea that after a mass extinction the mutation rate might be higher is rather natural. The environment has radically changed. This change may cause that the mechanisms that normally protect against mutations do not work as well as they should. Thus, there can be more mutations before the organism adapts to the new environment. But there is a problem.
The problem with a higher mutation rate is that mutations are mostly harmful. We can increase the mutation rate by e.g. radioactive radiation. The result is that the organism dies. Many of these harmful mutations develop into cancers. Some cause other problems. Increasing the mutation rate is likely to kill the whole population before any new useful gene has been created. The mutations should be limited to sex cells in order to protect the parents against harmful changes. It is also clear that female sex cells are too few and not under competition: not increased mutations should happen with the egg cells.
This leads to the solution: let there be more mutations in sperm cells. There are many sperm cells and they compete for access to the egg. Mutated sperm cells are usually not fit enough and will lose the competition, but some very few can be fit. These very few mutated cells can bring new genes to the gene pool. The sperm population is quite large and in this way this mechanism may solve the probability problem.
What should we see if this is the solution? Firstly, in the critical time when a new species is created, sperm cells would have more mutations than is normal. That would show as poor quality sperm as most of these mutations are harmful. The time of getting pregnant would increase as only a smaller proportion of sperm genes are fit. This is exactly what we see today in the developed countries where the environment for humans has radically changed from what it used to be. We should also see changes in boys. The most visible changes would be mutations in the X chromosome: as women have two X chromosomes and men have only one, there should be four times as many instances of changes in men than in women. This is exactly what is seen with Autism spectrum diseases. With Autism spectrum diseases there is a strange issue that high-performing, basically Asperger, type problems are more common with two European parents, while worse forms of Autism are more common with immigrant parents. It is unlikely to find changes in Y chromosomes because they correspond directly to sperm quality and unfit sperm cells lose the competition for an egg.
Considering this, I conclude that we live in a time of changes and can expect a major mutation in humans in the near future (say, a million years), provided that the modern style of life continues. Worsening of the sperm quality is not a problem, it is the solution.
There is one more thing I want to point out. Proposing a mechanism where only the mutation rate of sperm cells is increased means proposing an intelligent mechanism. It does not fit well to the Enlightenment time mechanistic materialism that impregnates darwinism and modern evolution theory. But that is simply a problem of that theory. We know that there is something like consciousness and intelligence and everything is not described by matter, because we ourselves have consciousness and other mammals also have it. It is idiotic to create a theory that ignores the fact that there is this concept of consciousness and to try to formulate a theory that only works with stupid mechanistic mechanisms. Why should it be stupid and mechanistic? Even quantum physics is not mechanistic. The only answer is that it is a religious believe: everything must be mechanistic and only matter exists. But it is not so.
It is always so with evolution theory. Everything has to be mechanistic and material, because otherwise comes the bible bogeymen from the bloody Yahweh religion with human and animal sacrifices. And if we have to ignore a known part of the reality, that is, the fact that there does exist consciousness, then it is worth it for fighting with those loonies. And we defend the good old chap Darwin and the Holy Science. But this is an old fashioned Enlightenment time free thinking, not for our times and not for science.
It is naive to think that such a simple mechanism as natural selection would be the only or even the main mechanism of evolution. It is most probably more difficult than that. We can implement natural selection in evolutionary algorithms and their achievements are not impressive. A live, conscious, and in the human case, thinking, organism reacts to changes in the environment in certain ways, often in intelligent ways. If the economic situation of the mother is poor, more girls are born. This is not a conscious decision, but it is an intelligent one. It is quite possible that in a changed environment the sperm producing cells go to a different mode and allow more mutations with the intelligent, though unconscious, intention of finding a way out of this problem by developing a better brain. In humans it was the brain that developed, so it would still be the brain that develops. It hardly leads to Asperger people. They are likely to be failed efforts, but it leads somewhere, given time, of course.