In the short run
The human evolution in the short run relates to the changes, accidental or not, affecting in works of SF of the isolated characters or many whole populations.
In this last case, one thinks initially of the nuclear or cosmic radiations which undergo the unhappy survivors of a Third World war or of an ecological catastrophe caused by the accidental meeting of the Earth and of a wandering asteroid.
In fact, the evolution already started! Without needing to unchain the apocalypse to upset the genetic inheritance of surviving humanity, some authors put in scene characters presenting as of the birth of the changes which make them, if not of the supermen, at least of the be separate.
The internal ear, of Robert Silverberg, introduces a telepathic hero, David Selig to us. This one does not appreciate really its capacity to perceive the thoughts of its similar, and culpability feels some. During the novel, as the character ages, its capacity grows blurred.
In more than human, of Theodore Sturgeon, four children with the exceptional psychic talents, considered as handicapped people, meet around an young man who lives with the variation of civilization. These five characters, who separate would have had few chances to survive: a child able to move the remote objects, two small binoculars which can appear or disappear, a baby mongolien with extraordinary capacities of calculation, and the young solitary adult, will form an entity capable of all the exploits.
Among the supermen of the Marvel comics, many which is those acquired their capacities following an accident having involved a change: one thinks well-sure of Spiderman (bite of mutant spider), in Incredible Hulk (exposure to the gamma rays), in X-Men (in the case of these supermen, the capacities are rather latent that accidental).
Seth Brundle, the unhappy scientist of film the Fly, David Cronenberg, carries out the first tests of a machine being used for téléporter the matter. Without the knowledge, it is locked up with a fly before carrying out a test on itself. After its transfer, Seth Brundle feels renovated and reinforced. What it takes for a spectacular improvement of its person is only the first stage of a terrible process of transformation, initiated by the fusion of its organization with that of a fly.