Medical emergency

A medical emergency was a situation in which someone was in immediate danger from an illness or disease with the inherent possibility of death.

In 2155, in the mirror universe, Doctor Phlox was summoned to the to deal with an unspecified medical emergency. It was, in fact, a rouse by $𝜇$ and $𝜇$ to allow them to speak with him and convince him to join their cause. 

In 2367, Lieutenant Commander Geordi La Forge declared a medical emergency before beaming up to the USS Enterprise (NCC-1701-D) with Lieutenant Commander Susanna Leijten, who was suffering from an alien retro-DNA virus. 

In 2369, Doctor Julian Bashir and Chief Miles O'Brien were called to an unexplained medical emergency on Bajor. In the end, it turned out to be the Dal'Rok, a mythical energy creature that manifested itself through an Orb. 

Later the same year, Aamin Marritza (posing as Gul Darhe'el) arrived on Deep Space 9 declaring a medical emergency while suffering from the later stages of Kalla-Nohra Syndrome. 

In 2370, while waiting in line at the replimat, Garak jokingly asked Doctor Bashir to declare a medical emergency in order to get them to the front.