Leg strength
Think you have strong legs? Until you can comfortably pedal to the top of the biggest, steepest climb in your area on a single-speed, you cannot truly claim to have thighs of steel.
Singe-speeds add a new challenge to your local climbs. The steeper it gets, the harder it gets and the options you have are to mash the pedals as hard as you can or admit defeat, get off and push.
With time, forcing yourself to push hard up the hills will make you a stronger rider, making hill climbs easier when you go back to your regular geared bike.
Efficient pedalling
When you are not going uphill, you are most likely spinning fast, wishing you could shift up a gear. This fast spinning trains a smooth cadence and builds an efficient pedal stroke over time.
Hit loose climbs and you'll be forced to keep a smooth consistent pedal stroke so as not to break traction at the rear tyre. Take this technique back to your geared bike and you'll be amazed at what climbs you can conquer.
In addition to teaching you to spin more efficiently, the bike itself is more efficient too with a direct chainline and no derailleur pulley wheels to soak up your precious watts. One pedal stroke on a single-speed and you'll immediately feel that direct attachment to the rear wheel.
Better mountain biking
If you slow down, you will have to pedal hard to speed up again. When acceleration requires extra effort, you learn to keep your momentum as much as possible.
Singlespeed mountain bikes encourage you to carry greater speed through corners or over obstacles. As a result, your riding will become smoother and more efficient.
Speed (sometimes)
If the conditions are right, a single-speed can be a very fast machine. On a rolling landscape with short climbs a single-speed rider can find themselves in a state of being constantly in a slightly too high gear. Consequently, you'll be pushing hard and fast to stay on top of that gear.
On flowing singletrail, a single-speed mountain biker will be hard to catch. You may fatigue early at first, but in the long run, you will end up far fitter.
Weight
A single-speed does not have a cassette, derailleurs, cables or shifters, and as a result, will weigh less than a bike with gears. A lighter bike may seem useless if you will end up spinning out anyway but it will accelerate faster out of corners.
Simplicity
And perhaps the biggest benefit, it's just simple. With no gears to index or cable tension to adjust, all you have to worry about is pedalling and braking. If you want to speed up, pedal faster.
And with such simplicity, it's cheaper too. Fewer parts are needed at first to set up the bike, and once riding, it's extremely reliable. Riding a single-speed, especially in winter, means that you do not have any gears to clean, no derailleurs to break and there are far fewer expensive parts getting ground away by grit.
Glory
Ok, so this can be achieved on any bike, but there's little better than beating your mates to the top of the climb without any additional gears. If you're the type to gloat about your fitness, this is very much your moment to shine.