The vermiform appendix was a part of the large intestine of Humans. Largely useless, most people disregarded its existence.
In 2264, Elias Sandoval's appendix grew back when he was exposed to the Omicron spores, the organ having previously been removed. (TOS: "This Side of Paradise")
In 2267, after Spock regained his sight, having previously thought he would be permanently blind, compared his Vulcan inner eyelid which had instinctively protected him to the Human appendix, in that both were generally ignored. (TOS: "Operation -- Annihilate!")
In 2377, B'Elanna Torres asked The Doctor to perform prenatal gene re-sequencing of her and Tom Paris's daughter to remove evidence of her Klingon heritage. Using the example of the baby's redundant third lung, she suggested that eliminating it would be no different than removing an appendix before it developed inflammation. (VOY: "Lineage")