What conditions should be met to notice a Doppler effect on light? What relativistic conditions should an observer meet to notice the Doppler effect on a sample light wave, and is it even possible for one?
 A: "Relativistic" velocities (velocities in excess of 0.1 c) are not needed. Any velocity difference will do. People use the doppler effect right here on Earth. Some sample uses:

Catching speeders.


(source: milb.com)
How fast is his fastball? (Very important at this time of year.)


Where is that tornado going?
A: As you clearly ask for a detection criterion of the doppler shift on "a sample light wave", I have taken the liberty of assuming the following scenario. You detect light from a source, emitting light of a known wavelength (the usual case in astronomy for instance) with an instrument that is capable of measuring wavelength with an accuracy (i.e. with an uncertainty) $\Delta \lambda$.
If you want to claim that you have detected a redshift with a single measurement, then usually you would have to see a shift that is at least 3 times the uncertainty in your measurement. i.e. $3\Delta \lambda$ (this is a very standard criterion, corresponding to a $>99$% confidence, but you can use whatever multiplier you feel comfortable with).
A doppler shift of this size will be measurable if you are moving at a relative speed $v$ with respect to the source of
$$ \frac{v}{c}  >  \frac{3\Delta \lambda}{\lambda},$$
where I am assuming that $v \ll c$ in order to use the approximate doppler shift formula.
If your instrument measures frequency instead, then you just replace $\Delta \lambda/\lambda$ with $\Delta f/f$.
If you are using a radar gun (that isn't in your question, but appears to be what others think you asked) then the doppler shift is multiplied by two because the shift occurs once upon reception by the reflecting object and then again when you receive the signal again. In this case, divide the right hand side by 2.
If your question refers to the human eye. Then according to this you are capable of detecting wavelength shifts of $\Delta \lambda \sim 1$ nm in ideal circumstances, which with my formula yields velocities of $\sim 2250$ km/s. (But this assumes you have a stationary, identical light source to compare the colour with.)
In astronomy, doppler shift measurements are becoming accurate enough to discern relative motions of just less than 1m/s.
The precision of earth-based measurements can be higher because often you are simply comparing an outgoing signal with a reflected signal. e.g. Radar guns are good to roughly $\pm 0.4$ m/s, but laser doppler velocimetry appears to be able to detect velociites as small as 0.2mm/s. http://www.usa.canon.com/CUSA/assets/app/pdf/encoder/LaserDopplerVelocitySensor.pdf
A: Your choice of the word "notice" is ill-defined. The Doppler effect holds true for all waves, at all speeds, so in that sense, you could "notice" the Doppler Effect for any wave you choose, no matter what the speed of the source or observer is (as long as there is some relative motion between the two.) Even as you walk across a room towards a lamp (to keep things simple, let's keep the light coherent) the light would be blue-shifted.  However, the change in apparent frequency would be so small that you would never be able to tell the shift on your own.
What you likely mean, though, is to be able to notice the effect with your own eyes without any other kind of measuring aid.  Human perception can be a tricky thing, but generally to notice the Doppler Effect in that sense, the speed of the source or observer has to be on the same order of magnitude as the wave in that medium.  The speed of light is huge ($c\approx 3 \times 10^8 m/s$) so for you to notice any apparent shift in frequency, either the source or observer should be traveling some measurable percentage of the speed of light.  This is most noticeable on cosmic scales where galaxies are usually moving away from (but in rare cases, moving towards) us at great speeds, and as a result, entire galaxies are red-shifted in the case of receding galaxies, or blue-shifted in the case of approaching galaxies.
As a semi-humorous afterthought, I've also heard the joke that cars on the highway are a perfect example of the Doppler Effect in light.  A stationary car has white headlights, but as a car drives away from you, its tail lights are red.  This is, of course, in tended to be humorous.  Highway speeds are far too slow, in the grand scheme of things, to notice a red-shift.
