We use Gaia Data Release 2 to search for possible surviving binary companions to three of the best studied historical Milky Way core-collapse supernovae. Consistent with previous work, we find there to be no plausible binary companion to either the Crab or Cas A supernovae. For the first time, we present a systematic search for a former companion to the Vela supernova, and rule out essentially any surviving luminous ($>L_odot$) companion. Based on parallax and proper motion, we identify a faint source (Star A; Gaia Source ID 5521955992667891584) which is kinematically consistent with being a former binary companion to the Vela SN progenitor. However, the inferred absolute magnitude of this source is extremely faint, raising the possibility that it may in fact be a background interloper. In addition, we derive a new distance ($3.37^{+4.04}_{-0.97}$ kpc) to the Crab SN based on the Gaia parallax measurements, which is significantly further than the 2 kpc distance typically adopted. Finally, we demonstrate that Gaia can be used to measure the secular decline in the luminosity of the Crab pulsar, and provide a new test of pulsar models.
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