Admittedly I've not dug into Thunderbird all that much. It downloads mail for me which is 99% spam which I delete.
When mail comes down, a icon is put in the sys tray. If I restore the Tbird window, delete the spam and then minimize it, the icon stays. It seems that I have to change mail folders then it removes the Icon. A minor annoyance, but, hey, is there some way to fix this? A stupid question for the weekend....;-) Steve