Is the light from the stars within our Galaxy redshifted? I know that metric expansion does not happen within a galaxy. But is the light from stars in our Galaxy somehow redshifted?
Also, everyone says that our Galaxy isn't expanding. But is there a slight chance, or has it been observed, that there is a very slight expansion within our Galaxy itself due to some unknown reason?
 A: Stars in the Milky Way do not have any cosmological redshift, as the entire galaxy is gravitationally bound and hence does not expand.
However, stars in the Milky Way can exhibit red as well as blueshifts. This is caused by the motion of the stars themselves, relative to the earth, as well as by the movement of the earth in its orbit around the sun. While these are only slow speeds (in cosmological terms) they are easily measured.
Even smaller shifts are observed when using the doppler effect for planet hunting. As a planet orbits its star, the star moves slightly: both star and planet are actually orbiting around the centre of mass of the system. These shifts are extremely small - the speed of the star is measured in m/s, compared to the speed of light of 300,000 km/s - but it is still measurable.
A: hdhondt's answer is perfectly correct. We do see systematic redshifts and blueshifts in various directions and at various distances in the Galaxy, albeit with large dispersions.
These systematics are largely connected with the Sun's own peculiar motion with respect to the average star in the solar vicinity. On this basis (and also by direct measurement of the observed tangential motion of Galactic centre radio sources) we think the Sun moves around the Galaxy at around 230 km/s, but is also currently moving inwards at a few km/s.
Also, different populations of stars have very different characteristics. Local Population I stars, which tend to be younger and have higher metallicity, generally display lower velocities (up to tens of km/s) with respect to the Sun, because they are also travelling on roughly circular orbits, similar to that of the Sun's, around the Galaxy. On the other hand, old Population II, low-metallicity objects travel on more spherical or even radial orbits around the Galactic centre and therefore have higher velocities with respect to the Sun - 100-300 km/s.
A secondary point is that the above discussion applies to systemic velocities. A large fraction of stars are actually in multiple systems and orbit their common centre of mass. This gives an additional Doppler signature that varies from almost undetectable for low-mass companions with wide separations, to of order 100 km/s for equal mass components in binaries with orbital periods of a day or so, or even higher for binaries involving compact stars (white dwarfs and neutron stars). These velocities of course vary on the orbital period.
An additional point is that all light emitted from stars is subject to a gravitational redshift. This is generally smaller than the above motional effects - of order 0.5 km/s for a star like the Sun. However for more condensed white dwarf stars the effects can be as large as 50-100 km/s.
