In short:
You cannot mount an EF-S lens to an EF body unless you do some hardware modifications. If you mod it, then yes, it will mount and it will work, but you might get vignetting and loss of corner sharpness.
An APS-C lens with an EF mount (Sigmas, Tamrons, Tokinas) will mount and work on an FF body without any modifications. However, they may exhibit vignetting at some or all of their zoom ranges, depending on the lens.
Some APS-C lenses do work pretty well on FF or 1.3x, though. I've heard of quite a few people using the Sigma 10-20mm as an ultra-wide on their 1D series bodies.
With Nikon, a DX lens will work on an FX body, and you have the option for automatic in-camera cropping to the DX sensor size.