Are there concepts of mathematics today that aren't on a solid logical foundation? In this answer, Arturo wrote:

I would think like Sebastian, that most "working mathematicians" didn't worry too much about Russell's paradox; much like they didn't worry too much about the fact that Calculus was not, originally, on solid logical foundation. Mathematics clearly worked, and the occasional antinomy or paradox was likely not a matter of interest or concern.

And I wondered: is there some mathematical concept today that is not in a solid logical foundation?
 A: Generally there always exists a framework that gives solid logical ground (modulo inconsistency issues with the framework).  But often this framework is a little more restrictive and a requires mathematicians to be a little more careful then they, in practice, are.  So, while one can claim that such foundations exist the claim that mathematicians actually use these foundations is pretty sketchy.
For example most mathematicians would say that ZFC puts set theory, and hence pretty much the rest of math, on solid logical foundations.  But there are issues with using ZFC to do category theory.  Most mathematicians do not care if their categories are small and make no attempt to address this issue.  Also the definition of a derived functor, for example, is usually presented in a way that is not quite correct if you use ZFC as your foundations.
Both of these issues can be fixed by taking ZFCU as your foundations, but this just pushes the laziness issues a bit farther down the road.  For example, most mathematicians don't bother to check that what they are doing is independent of a choice of Grothendieck universe.  They assume it is, and are generally right.  But there do exist examples where this isn't the case.
A: Well, we don't know whether $\mathsf{ZFC}$ is consistent, so I guess modern set theory is on pretty shaky ground!
