Final answer:
The most likely mode of transmission for the trait described is rare X-linked recessive, (option 4) as it is expressed in half of the sons and not in the daughters nor the parents, indicating a sex-linked pattern where the trait is carried on the X chromosome.
Step-by-step explanation:
The trait in question is most likely inherited through a rare X-linked recessive mode of transmission. This is because the trait is not expressed in the parents, but is observed in half of the sons. If the trait were autosomal recessive or dominant, then daughters would also have a significant chance of expressing the trait depending on the genotype of the parents.
However, the specificity of affectation to sons indicates a sex-linked pattern, and because sons inherit their only X chromosome from their mother, the trait would only be transmitted via this X chromosome. Given that daughters are unaffected, it suggests that the trait is not dominant (which would affect both sexes more uniformly), but rather recessive and carried on the X chromosome. This pattern of inheritance aligns with the way in which X-linked recessive traits operate in human genetics, where females can be carriers without expressing the phenotype, but males who inherit the recessive X-linked allele from their carrier mother will express the trait.