Examples of non-sine waves? What would be a non-sine wave? AFAIK, all sound is a sine wave, equally to waves on the sea. What would be a common example of something in nature that's a wave but not a sine wave? Or, would we have to look at man made regularities like bus timetables or stock prices to find non-sine waves?
 A: Vowels, for example. From this page the waveform of a female voice (frequency about 250 hertz):

Then there is Fourier's theorem that any (analytic) periodic signal can be written as a sum of harmonic sine waves.
A: Waves can take all sorts of funny shapes. This is especially true if the differential equation generating them is nonlinear. This makes fluid mechanics a wonderful source of complicated waves. My personal favourite example is the solutions $\phi(x,\,t)=-\frac{c}{2}\operatorname{sech}^2\left[\frac{\sqrt{c}}{2}(x-ct-a)\right]$ of the KdV equation. See here, here and here for more examples.
A: It is not true that all sound is sine waves.  What is true is that one can find sinusoidal solutions to the wave equation.  This allows an arbitrary wave to be expresses as a linear superposition of sines.  It looks like the Soliton has been mentioned above.  While this is one counter example it relies on the non-linear KdV equation.  I want to point out that even the linear wave equation allows for strange non-sine looking solutions.  The decomposition of an arbitrary wave or pulse in terms of sines is related to the Fourier transform (also mentioned).  
