I know there's a bit of a fad for both single-speed conversions and "fixed gear" bikes.... We have active threads on these at Bikeforums.
I see a lot of the single-speed conversions here at the university; it's pretty easy to take an old road frame with horizontal dropouts and install a single-speed "kit".
Advantages... It's simple, light, reliable. Hop on the bike and ride. If you have chosen a gear ratio that gets you around where you ride, you're golden. (the university is pretty flat, for the most part)
Disadvantages... Hills. If you are young, strong, and/or light, then you can muscle your way up a decent hill by either mashing or standing.
However.... If you are like me... Old, hefty (200-ish) and with wonky knees, then gears are your friend. I LOVE gears. The more the better, thanks. I'm a mechanic so my shifting is always spot-on and crisp, and I do it constantly and without thinking.
Tullio Campagnolo did not invent the derailleur for no reason.
Fixed-gear. What can I say? For many years, the paradigm for the pro road racer was to spend a portion of Spring (early-season) training on a fixed-gear bike to develop "spin".
Of course, if one puts the bike in a lower gear and simply doesn't shift... You accomplish the same thing.
As far as I know, this traditional training method has gone by the boards.
For the enthusiast, I suppose there might be some charm in learning to ride a bike similar to what our ancestors used. Even going so far as (alluded to above) to have no brakes.
For those not familiar, this means that the fixie rider must either backpedal (apply back-pressure) to stop, or apply the soles of one's feet to the tire....
A hazardous undertaking at best.
It was not at all unusual for our ancestors to "loose the pedal" when going downhill, which almost always resulted in a severe crash.
Again, I suppose there might be some charm to mastering these arts; much as with riding a unicycle or "giraffe" bicycle.
However, as a practical form of transportation, we again note that such inventions as freewheels and efficient brakes were enthusiastically embraced by the cycling world.