The only time I wear an ESD strap is when I'm working on flight hardware, in the clean room, or I am presenting or teaching. Those latter two I only do because if I didn't I'd have people getting all anal about it and not concentrating on what I've come to talk about. On flight hardware and in the clean room it's as much psychological as anything else, it's part of a regime to take care in general.
I was working on a project a couple of months ago in Dubai and we had a visit from almost the entire Dubai and UAE governments, including the two top dudes, the ruler of Dubai as well as the President of the UAE. We had one of our engineering model satellites out demonstrating its stuff, and one cabinet minister came over and started prodding it, much to the chagrin of the poor young engineer I'd been working with! I looked from afar and maintained a wry smile.
Have I ever had a part fail where I could say it was due to ESD? Personally speaking, no. I have, however, had products returned to me where, for example, the USB peripheral has gone (protected, I might add, with external protection diodes). Whether this was ESD, or a bad host port, or a transient of some sort, I don't know, but it does happen very occasionally.
Do I take care with parts? Yes, they're generally kept storedin their reels/strips/tray until needed, and then inside metallised zip lock bags. Do I use an earthed ESD mat? Yes. Do I take care to earth myself and the board I'm working on to the ESD mat? Not particularly, but having the mat there means it's difficult for myself, the board and the irons not to be at the same potential. But the ESD strap only goes on occasionally.