What is the most likely mode of transmission for a trait that is not expressed in parents but is expressed by one half of the sons?

Answer: Rare X-linked recessive

Explanation:

In humans our biological sex-determination is based on a pair of sex chromosomes namely, XX in females, XY in males. The genes on our X chromosome are X-linked. and have special patterns of inheritance due to their presence in contrasting numbers in males (XY, 1 X chromosome) and females (XX, 2 X chromosomes); thus they are more common in genetic males.

The observed trait is X-linked, and given that only half the sons are affected, the trait is recessive and carried by the heterozygous mother - heterozygous alleles are two variants of the allele at the same gene locus. The Y chromosome inherited from the father does not carry the gene, thus there is no normal copy available. The incidence for its inheritance is at 50% as the mother has two X hromosomes, one normal, and one carrying the trait.