Single-year spikes in radiocarbon production are caused by intense bursts of radiation from space. Supernovae emit both high-energy particle and electromagnetic radiation, but it is the latter that is most likely to strike the atmosphere all at once and cause a surge in 14C production. In the 1990s, it was claimed that the supernova in 1006 CE produced exactly this effect. With the 14C spikes in the years 775 and 994 CE now attributed to extreme solar events, attention has returned to the question of whether historical supernovae are indeed detectable using annual 14C measurements. Here, we combine new and existing measurements over six documented and putative supernovae, and conclude that no such astrophysical event has yet left a distinct imprint on the past atmospheric 14C record.