Is there a candidate for a natural problem in $P/poly - P$? I want to know if non-uniformity helps computing functions in practice. It is easy to show that there are functions in $P/poly - P$, take any uncomputable function $f$ and consider the language {$0^{f(n)}:n\in \omega$}, which clearly has simple non-uniform circuits, but is not computable uniformly at all, but this is not the kind of functions I am interested in.

Is there a function that we know it can be computed non-uniformly but we don't know if it can be computed uniformly (or at least proving that it can not be computed uniformly is not obvious)?
How can non-uniformity of circuits be used for computing functions that are not known to be computable uniformly (with almost the same amount of resources)?

Please note that I don't want pathological functions like uncomputable one mentioned above, I want natural functions that people are really interested in computing and it is plausible that can be or could have been computed uniformly.
Edit:
I know that $BPP \subseteq P/poly$. So an answer which is not a derandomizations result is more interesting for me.
Edit 2:
As András Salamon and Tsuyoshi Ito have said in their answers, $Sparse \subset P/poly$, and there are interesting problems in $Sparse$ that are not known to be in $P$, so formally they have answered what I have asked, but that does not help with what I am really interested in since the reason that they are in $P/poly$ is the possibility of hard coding a sparse language into the circuit. A language which is not sparse would be more interesting.
I do not know if this satisfies your requirements, but Bill Gasarch’s blog post in July 2010 asks about languages in SPARSE∩NP which are not thought to be in P, giving an example from Ramsey Theory.  Any such languages belong to (P/poly)∩NP.

Related to this, For any language L∈NP, the language TL={1n: L contains some string of length n} is in TALLY∩NP ⊆ SPARSE∩NP ⊆ (P/poly)∩NP.  Depending on the choice of the language L, TL may not have an obvious reason to belong to P.