Figure 1. Source.
Figure 2. Distance of stars nearest to the Sun as function of time. Source.
Some notable ones:
  • Proxima Centauri, the nearest one, at 4 ly. It is part of the Alpha Centauri star system, which contains two other stars at very similar distances as well, and their relative distances to earth will change positions in a few tens of thousands of years.
  • Sirius, the brightest star in the sky at 9 ly