I use a Korg GT-12. Apparently it has been discontinued and replaced with the GT 120, although appearance-wise they are quite different.
I like it a lot better than the GA-30 that I still keep in my bass bag as a backup. Of course it cost like 4x as much.
I do not ever run it as part of my signal chain. I just use it for checking pitch on open strings before practicing, before a gig, and sometimes between sets. Running the cable from the pickup to the input works much better than the built-in mic, unless I am in my music room at home. Maybe that has to do with the harmonics and overtones mentioned above.
And like others have noted, many of the gigs I do are w/ a piano (i.e. not an electric keyboard), which is more often than not never exactly at A=440Hz. So my "perfectly tuned" open strings are then out of tune ... I have thought of tuning up or down to match the piano, but that can affect string tension, which will bother me if it is too far out of whack.
I think flatback mentioned using the clip-on variety attached to the bridge for practicing, esp. working on thumb position, which I thought was a great idea.