On one of the forums, I suspect that there is a recurring troll. I cannot prove this, since I don't know who is behind the messages. What is the best thing to do - nothing, or report the messages and ask moderators to look at it?

I would say report him, but try to show the comments to support, I.e. give a link or screen shot. Past that just ignore the troll. They will go away if they do not get responded to.