How Neutron Stars Get Their Kicks
mustermark writes: "Now we may know why neutron stars zip through space at 1000 times the speed of a normal star. Massive stars have been shown to collapse aspherically, and chunks blown off in this process may recoil the neutron-star remnant in the opposite direction."
Since there aren't really any signs in the Universe that say "I am perfectly stationary, measure from me"
Actually, there is a pretty good one - the microwave background radiation.
There is a nice picture here which shows the relative temperature of this radiation as seen from earth. It is clearly red-shifted in one direction and blue-shifted in the opposite direction, indicating that the earth is moving rapidly (600 km/s) in the direction of the blue-shift. Some of this velocity is our motion around the sun, some is the sun's motion in our galaxy, but most of the velocity is common to our entire local group of galaxies.
If we were "at rest" in the universe, the microwave background would be uniform in all directions [more precisely, the dipole component would be zero; quadrupole and higher terms would still be present]. However, distant stars and galaxies would still be moving away from us due to the expansion of the universe.