Wall power socket outlet: square or sine wave?
Utility power generators produce three-phase sine power.  One reason for this is that the power is constant within a cycle.  That means there is no torque ripple on the generator shaft.  If there was, these large machines would be vibrating themselves into pieces.

Another reason for sine waves is that there is no harmonic content.  If the generator puts out 60 Hz, then 60 Hz is the highest frequency that the rest of the transmission system has to deal with.  Any other waveshape than sine would contain other frequencies.  This would complicate parts of the transmission system, especially the transformers.

That said, while the power voltage is ideally sinusoidal, in practice it's not perfect, of course.  Most of the reason is that loads don't always draw sinusoidal current.  Think of a full wave bridge, for example.  It draws current in spikes at the tips of the waves.  Since the impedance of the power line isn't zero, this non-sinusoidal current draw causes some non-sinusoidal voltage.  Since non-sinusoidal voltage is undesirable, there are regulations about how much the current draw is allowed to vary from sinusoidal for high power loads.