Not too much, actually. People hanging up without a message are the most common. There are also a few people that have played recordings of songs or other sound clips from copyrighted sources, and I don't want to deal with that mess if I don't have to. There are also the ones that are in other languages that I can't put a good memo on or moderate properly. I keep them all and just leave the ones that I don't post marked as "unpublished" in my static site builder (Jekyll) so that I can take another pass at them later when I have the time / resources.
Very few have been "inappropriate". There was one with some questionable jokes that I held back and another where someone was quite angrily talking about a particular political party (there are other "political" statements I've allowed; I just don't want to the page to become some place for people to yell at each other -- not going for that vibe).