Wildly, wildly, wildly, wildly useful. SHM will show up over and over and over again in physics in various forms other than actual, physical springs. A couple of specific examples are: simple pendulum motion, a positive charge given a small kick at the center of a ring of uniform positive charge density will undergo simple harmonic motion, after some idealizations if you drill a hole in the earth and drop down you will execute SHM, and a marble rolling in a circular dish in the small angle approximation will undergo SHM. There are so many rich examples of SHM out there; you will find a plethora of examples on the internet. By showing the object undergoes SHM through newton's 2nd law you can read off the proportionality constant from the DE.