How commonly supported is the Stencil Buffer right now by hardware in OpenGL?
Stencil buffers have been commonly supported by GPU since ancient times.
Is it better/worse supported than shaders?
"Is it colder than outside at night?" Yeah, that question makes about as much sense. Stencil buffers server a completely different purpose than shaders.
If the hardware doesn't support it, will it be emulated in software?
Yes, but not in a desireable way: The whole render pipeline will drop back to pure software rendering. The stencil test is implemented by hardwired functionality and can not be emulated by a "drop-in" software that only implements the stencil test while the rest was implemented differently. It simply doesn't work that way.
Luckily every GPU you'll encounter in the wild does support stencil tests. So this is of no concern whatsoever.