Is there any advantage to soft pedaling over coasting? Does it provide any power to moving forward? Is there any aid it gives to your leg muscles recovering?
By this do you mean pedalling when unnecessary (and without increasing the bike's speed), for example, on descents?
I will do this on long descents just to keep my legs turning over and to prevent them getting too cold. But I will mix it with coasting.
There is a theory that keeping the legs moving will keep your blood flowing and help pump lactic acid out of the muscles. I can go along with this as I've felt the benefit, or at least I think I have.
I've also heard it argued that it's advantageous as regards making your technique smoother, but I don't think that's particularly proven. But no personal experience, I've never really concentrated on my technique to that extent.
Further, you can talk to racing cyclists who will often say that soft pedalling is preferred when riding in a peloton, purely because coasting implies to other people that you're about to hit the brakes. Again, no personal experience. I know some of the regulars on here are either racers or ex-racers, maybe they can confirm?
In terms of speed (or power) surely it makes no difference, purely by the definition of "soft pedalling"?
In addition to the response provided by PeteH, I use "soft pedaling" when coasting to a stop on my downtube shifter/derailleur-equipped road bike in order to change gears before a stop.
Some bikes (such as those with internal hubs) do not need any chain motion to shift gears, but my bike does. That's the only concrete reason why I would employ this technique, outside of the points outlined by PeteH.
When riding in a group you often find that small changes of speed can mean that you do not need to pedal when it slows down and have small bursts of power when it speeds up. By soft pedaling when the group slows down your legs will already be spinning when its speeds up again.
I have often heard, and I agree, that the "muscular pump" is partly responsible for higher blood flow to specific muscle regions during activity. The reason is that between muscular contractions the difference in pressure is such as to draw blood from the arterial side to the venous side. It is a hypothesis and has not been conclusively proved. That may be the reason why some athletes do it, and others don't.
- Cassey, Darren; Hunt, Emma (2008), "Cardiovascular function in humans during exercise: role of the muscle pump"
- The muscle pump raises muscle blood flow during locomotion.
- Clifford PS, Hamann JJ, Valic Z, Buckwalter JB. Counterpoint: The muscle pump is not an important determinant of muscle blood flow during exercise. J Appl Physiol. 2005 Jul;99(1):372-4
Although I'm sure it's not the same, in MTB I use a very brief lapse of smooth pedalling while downshifting in a difficult ascent, it helps reducing noise and wear in the gearset. I accelerate a little before the shift, then, as I actuate the shifter I pedal without load relying on inertia. When I feel the new gear is fully engaged I resume normal pedalling. This is also used when after a fast descent comes a very steep ascent, as you would normally be in a very long (hard) gear, soft pedalling is used to make sure you have the proper (selected) gear engaged when you need it. It is specially useful when the bottom of this kind of transition has technical terrain that would make it impossible to keep all the momentum, forcing you to reduce speed a lot but not to stop nor dismount.
Just based on personal experience. No to all your questions.
Jacques Anquetil, when time trialing, was rumored to incorporate a soft pedal stroke every few revolutions. With his smooth pedaling and only archival vintage video it's hard to tell. Could be baloney like the special oxygenated recovery beverage the Soviet track cyclist used.