Which star's 2016 planet is the nearest possibly habitable world to Earth?

Proxima Centauri, 4.24 light years away, whose planet Proxima b orbits inside the zone where liquid water could exist. It was found by watching the star wobble - the planet's gravity tugging it back and forth by about a metre per second, roughly walking pace, detected across forty trillion kilometres.

Habitable zone is a narrow technical claim, and the situation is probably grim. Proxima is a red dwarf, so the zone sits very close in - the planet's year is 11 days - close enough that it is likely tidally locked with one face permanently lit. Worse, red dwarfs throw enormous flares: one observed in 2019 briefly brightened the star by a factor of a thousand, which over billions of years is a plausible way to strip an atmosphere entirely.

It is nonetheless the only interstellar target that looks reachable. Breakthrough Starshot proposes gram-scale probes on light sails pushed by ground-based lasers to a fifth of light speed, arriving in about twenty years - a serious engineering programme with several unsolved problems, aimed at this specific star because there is nothing closer.