If you go a little bit further than the inverse and implicit function theorems, you can get a fairly practical theorem. Kantorovich's theorem gives you fairly strong sufficient conditions for a system of smooth equations to have a solution. Moreover it tells you how quickly Newton's method converges in that situation. For example, this theorem is used by Harriet Moser to prove that SnapPea does give approximations to actual solutions to the hyperbolic gluing equations. The applications of course are pretty broad, this is one on the fairly pure end of the spectrum. Kantorovich was an economist although I do not understand the economics problems he was interested in.
If you're interested, this perspective on the inverse and implicit function theorems is in "full glory" in Hubbard's multi-variable calculus text.
2nd answer: The proof of Sard's theorem is a delicate dance with the Implicit Function Theorem, Taylor's Theorem and some basic argument with Lebesgue measure zero.
