in my experience the most common lens problems will be:
1. decentering (and note, out of all lenses i have tested the majority have some small variation between sides/corners, so this has to b within reason). this is very common.
2. different AF MA needed at different distances (this is a problem for Canon lenses since no body that I know of let's you use different MA values for different lenses yet, note that Sigma does allow calibration in this manner). this is rare but i do have lenses that do this.
3. different AF MA needed in different lighting conditions. unfortunately this is hard to tell if it's working as expected or something is broke. so, dunno if this is worth testing.
note, i don't consider checking general AF to be a "problem". about 1/3-1/2 of all lenses i have ever used needed some kind of MA and various bodies. it's just something you have to do. if you got something like a Rebel that doesn't have MA, well, sorry.
to test #1, you should take pictures of stuff really far away to take DOF out of the equation. also shoot at the biggest apertures in that case--they will be most revealing.
to test the rest, just shoot a variety of things.