It's even worse. They practically never get seen flying any more. In an interesting book, "Watch the Skies: A Chronicle of the Flying Saucer Myh", Curtis Peebles traces how the working myth of flying-saucer proponents evolved over the years, and examines how sightings changed with social factors. The last of the classic flying-saucer flaps happened around 1973; reported sightings changed from flying disks, etc. to reports of abductions and close encounters. (And when you come right down to it, the objects Kenneth Arnold described in 1947 weren't saucer-shaped, he just compared their motion to skipped saucers.)

He also cites work by Otto Billig pointing out that these flaps coincided well with periods of ill-defined tension at least in Western societies. The "ill-defined" part was important - the Cuban Missile Crisis (aka Caribbean Crisis aka October Crisis) didn't count, for example, because the reasons for dread were quite specific and well-defined.