1. 3 phase just seems to be the easiest and most economical way. If you look down the big powerlines, you will always see pairs of 3 lines. Generators naturally make 3 phase power. The frequency 50/60 is simply set by the generator speed. 60 hz is 3600 RPM I believe. Due to the poles in a generator, the generator spins at a different speed, more likely 1800 RPM for example. You could use a 6 phase or a 9 phase or whatever, its just that 3 phase just makes the most sense monetarily and so forth.
Perhaps you could say that the naked eye can't pick up 60 hz in lights or whatever, but I think It is set that speed more because of the speeds of the motors we use. Your AC condenser, blender, air compressor, house fan, etc...all turn at a nice speed because of this 60 hz. If the poles are set correctly, a generator spinning at 3600 RPM could nicely turn a motor at 3600 RPM. Thats nice harmony...generator and motor are almost identical.
3. Single phase simple pulls off of two of the line to line phases...or line to nuetral phase. Instead of three sin waves intertwined and shifted 120 degrees from each other, the single phase sine wave is just that...a SINGLE sin wave that repeats 60 times per second...or 60 hz.
3 phase is best for distribution, commercial and industrial applications. It just makes sense due to the massive power needed.
However, in residential homes, single phase works just fine. For example, say you need to turn a huge 200 HP motor in an INDUSTRIAL application that turns a huge water pump or something. This motor needs to last for 20 or 30 years. 3 phase would be best. Not only would single phase be akward, it would shake the bearings right out of that motor after a long time. 3 phase is much smoother. Picture three torque arms turning a motor...that's three phase...nice and balanced. Now picture one torque arm turning a motor. Not so smooth, definitelly not balanced.
That being said, single phase is fine for turning you can opener, blender, refriderator condenser, mictrowave, dryer and even your smallish AC condenser in your home.
USA uses 240/120 single phase system in homes. Places like UK and others do actually bring 3 phase into residential. 208/120 I believe.