Which is more mechanically efficient, a bicycle or a unicycle? Human physiology aside, I wonder which device is more mechanically efficient, a bicycle or a unicycle?
 A: According to Wikipedia:

A bicycle chain can be very energy efficient: one study reported efficiencies as high as 98.6%

Given that on a unicycle, there is no chain, it ought to heave fewer losses in the transmission system between foot and road.
However, if you look at overall efficiency of the whole human plus bicycle system for a specific activity, a conventional geared bicycle may be a more efficient means of transport due to gearing, the availability of brakes and other factors. Human pedalling efficiency is best when the person maintains an optimal pedalling cadence. This might be more difficult on a cycle that has no gears. At a minimum, the unicycle will have only one optimal speed where a geared cycle has many.
A: A bicycle is mechanically more efficient as a vehicle.
In the unicycle we can't harness the kinetic energy of the vehicle.  The unicycle is direct drive, no gears, no "coasting".  The system (unicycle and rider) requires continual addition and removal of energy (kinetic energy) to remain balanced.  The center of rider mass must be over the contact point of the wheel to avoid the common asphalt body scrub. Continual acceleration and deceleration of the wheel keeps it beneath you. 
The unicycle is mechanically less efficient due to the balance requirements.
I have ridden a unicycle for almost 50 years, started when I was in first grade.
A: This is not meant to be a me too answer. 
To do a just comparison we need to examine  the different vehicles at similar speeds and terrain. Terrain may be downhill, uphill or flat or uneven. We must also consider factors such as road surface, unevenness and the total motion of the system as it moves on differing surfaces.
There is a noticeable phenomenon I call wobulation that a climbing or descending uni rider exhibits.
Since the body of the uni rider is constantly striving to remain centered over the unicycle, and their is no "rudder" effect from a second wheel, much energy is expended to maintain ones line of travel. 
The unicycle at a posited top travelling speed of lets say 15 mph (29 or 36 incher designed for distance travel) has a tendency to gyrate first to the right, then to the left of the line of forward motion. This constant crossing and correcting on the center line wastes considerable energy and the unicycle can be seen deviating side to side several cm with each wheel revolution. Check out videos of any of the many long distance uni riders on Youtube.
There is math to support what is observed. I am an electrical engineer and unicyclist and not prepared to exhibit a math proof as I just came across this FAQ. Any serious cyclist who has ever ridden for much distance with a uni rider knows that for the same amount of payload and same terrain the bicyclist does lots of waiting and holding back because the uni rider is severely limited by the system ridden.
The unicyclist's wheel path is a very obvious wavy line, Bicycles exhibit this wobble behavior also but not to nearly the same degree because of the stabilization of the second wheel Note: an experienced cyclist in a tucked position can hold a steady line with both wheels tracking straight at some speed.
Unicyclists use energy in two planes plus slight vertical deviation to make progress. Bicycles use much less energy in the second plane and vertically.
Slim Chestnut,
Engineer for over 55 years. Bicyclist and unicycist. Cycling Instructor and advocate for cycling. I have over 150,000 miles of cycling experience.
The unicycist "Boberto" is correct about the extra work unicyclsits do.
A: I'd say a unicycle due to less friction. Though it would be harder to control.
My reasons as to why it'd have less friction are the same as @DIYser.
A: Unicycle.  Less friction (no chain, no rear sprocket, no rear tire & less overall weight).
