Human DNA
Vulcan DNA
Deoxyribonucleic acid (DNA) is the building block of life. It is the blueprint for a living organism and encodes amino acids for polypeptides which form proteins, which, when properly sequenced, form cellular pathways that together form the physical structure and anatomical characteristics of an organic lifeform.
DNA samples are collected from all Vulcan children at birth and kept on file by the Vulcan government. (ENT: "The Forge")
By matching DNA gene types, Starfleet was able to identify Jono as Jeremiah Rossa. (TNG: "Suddenly Human")
If DNA is changed in some way, the host organism responds to the change by re-making itself in the form specified by the new DNA, often extremely rapidly. Semi-corporeal beings such as the Founders also had DNA though there is no evidence they had any cellular structure or indeed any cells.
In 2369, following clues developed by his mentor, Richard Galen, Jean-Luc Picard discovered evidence that an ancient race of humanoids had seeded many planets with their DNA billions of years earlier. The evidence suggested that Humans, Romulans, Klingons, Cardassians, and others may have far more in common than previously believed. (TNG: "The Chase")
DNA extracted from a male could be used to impregnate a female. In 2372, Seska claimed she impregnated herself with DNA she had stolen from Commander Chakotay while he was her prisoner. (VOY: "Maneuvers")
The missile Cardassian ATR-4107 ("Dreadnought") used a DNA scan on anyone visiting it to establish their identity. (VOY: "Dreadnought")
In 2374, certain segments of the DNA of a number of USS Voyager crew were hyper-stimulated as part of the experiments by the Srivani, resulting in dangerous mutations. (VOY: "Scientific Method")
In 2375, Commander Valerie Archer (secretly a member of Species 8472 in disguise) shared a kiss with Commander Chakotay to obtain his DNA, which she was then able to analyze to prove that he was human. (VOY: "In the Flesh")
A strand of DNA, presumably from an Earth species was seen in the resetting time stream after the end of the Temporal Cold War. (ENT: "Storm Front, Part II")
Background
As DNA research was relatively new in the 1960s (DNA structure was first established in 1953 with the first notable DNA sequencing research conducted in 1973), Star Trek: The Original Series has no mention of this rather significant piece of medical science, even though DNA would have been well known in the 2260s according to Star Trek canon. It is also for this reason that Original Series episodes depended on other medical methods, whereas DNA samples would have been the preferred choice. Such examples include the voice print identification of Kodos, seen in TOS: "The Conscience of the King", and a reference to "medical plates" during the examination of Peter Kirk in TOS: "Operation -- Annihilate!".