I fly southwest all the time due to how cheap they are, I have never had any issues, but I had no idea they didn't overbook or have this reputation for CS either. Like, the fact that everyone else does this stuff while charging MORE for their flights seems ridiculous. Yes, southwest isn't like luxurious, but unless you're going on an insanely long flight (like out of the country) it's totally fine.
They stopped overbooking a month or so ago.