In addition to the the question in the title I'd like to know , how scientists reacted to the exceptions?
I would say that first exception might be White gene in Drosophila Melanogaster, described in 1910 by Thomas Hunt Morgan and Lilian Vaughan Morgan. At least, is the first exception we study in genetics since sex linkage its so fundamental. These experiments established the chromosomal theory of inheritance.
Thomas Hunt Morgan, an embryologist who had turned to research in heredity, in 1907 began to extensively breed the common fruit fly, Drosophila melanogaster. He hoped to discover large-scale mutations that would represent the emergence of new species. As it turned out, Morgan confirmed Mendelian laws of inheritance and the hypothesis that genes are located on chromosomes. He thereby inaugurated classical experimental genetics.
After breeding millions of Drosophila in his laboratory at Columbia University, in 1910 Morgan noticed one fruit fly with a distinctive characteristic: white eyes instead of red. He isolated this specimen and mated it to an ordinary red-eyed fly. Although the first generation of 1,237 offspring was all red-eyed but for three, white-eyed flies appeared in larger numbers in the second generation. Surprisingly, all white-eyed flies were male.
These results were suggestive for hypotheses of which Morgan himself was skeptical. He was at the time critical of the Mendelian theory of inheritance, mistrusted aspects of chromosomal theory, and did not believe that Darwin's concept of natural selection could account for the emergence of new species. But Morgan's discoveries with white- and red-eyed flies led him to reconsider each of these hypotheses.
In particular, Morgan began to entertain the possibility that association of eye color and sex in fruit flies had a physical and mechanistic basis in the chromosomes. The shape of one of Drosophila's four chromosome pairs was thought to be distinctive for sex determination. Males invariably possess the XY chromosome pair (Morgan used a more cumbersome notation) while flies with the XX chromosome are female. If the factor for eye color was located exclusively on the X chromosome, Morgan realized, Mendelian rules for inheritance of dominant and recessive traits could apply.
In brief, Morgan had discovered that eye color in Drosophila expressed a sex-linked trait. All first-generation offspring of a mutant white-eyed male and a normal red-eyed female would have red eyes because every chromosome pair would contain at least one copy of the X chromosome with the dominant trait. But half the females from this union would now possess a copy of the white-eyed male's recessive X chromosome. This chromosome would be transmitted, on average, to one-half of second-generation offspring—one-half of which would be male. Thus, second-generation offspring would include one-quarter with white eyes—and all of these would be male.
Intensive work led Morgan to discover more mutant traits—some two dozen between 1911 and 1914. With evidence drawn from cytology he was able to refine Mendelian laws and combine them with the theory—first suggested by Theodor Boveri and Walter Sutton—that the chromosomes carry hereditary information. In 1915, Morgan and his colleagues published The Mechanism of Mendelian Heredity. Its major tenets:
• Discrete pairs of factors located on chromosomes like beads on a string bear hereditary information. These factors—Morgan would soon call them genes—segregate in germ cells and combine during reproduction, essentially as predicted by Mendelian laws. However:
• Certain characteristics are sex-linked—that is, occur together because they arise on the same chromosome that determines gender. More generally:
• Other characteristics are also sometimes associated because, as paired chromosomes separate during germ cell development, genes proximate to one another tend to remain together. But sometimes, as a mechanistic consequence of reproduction, this linkage between genes is broken, allowing for new combinations of traits.
Morgan's experimental and theoretical work inaugurated research in genetics and promoted a revolution in biology. Evidence he adduced from embryology and cell theory pointed the way toward a synthesis of genetics with evolutionary theory. Morgan himself explored aspects of these developments in later work, including Evolution and Genetics published in 1925, and The Theory of the Gene in 1926. He received the Nobel Prize in Physiology or Medicine in 1933.
The first exception to Mendelian genetics was probably discovered by Mendel. As a followup to his pea experiments, Mendel used hawkweed, in which his rules do not seem to apply due to some unusual aspects of hawkweed development; see Apomixis in hawkweed: Mendel's experimental nemesis for details.
Mendel eventually abandoned his biological experiments, and though the main reason was probably his increasing other responsibilities, his thousands of apparently-unsuccessful experiments with hawkweed may have contributed.