A number theorist I know (who studies Galois representations) raised a question recently:



> 
> Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 classes where the character takes nonzero values? 
> 
> 
> 


He has learned about a few special examples involving nonabelian groups which are very close to being abelian. My limited intuition about the question, based on finite groups of Lie type, suggests that the sort of group he is looking for will be far from simple. But maybe there is no reasonable characterization of these groups for a given $n \leq 4$? In any case, there should be some relevant literature out there. The question itself belongs to a familiar genre in finite group theory: What does the character table tell me about the structure of a group? 
Appropriate references: S. , Gagola, Pacific J. Math 108 (1983), 363-385,


Berkovich-Zhmud', Characters of Finite Groups, Part 2, Chapter 21.



Finite groups with a character having very few nonzero values?
For the case of n=2 see Gagola's paper "Characters vanishing on all but two conjugacy classes" MR0721927. Some improvements on Gagola's results can be found in Kuisch and van der Waall's papers "Homogeneous character induction [I and II]" MR1172440/MR1302857 and in my paper ["Groups with a Character of Large Degree"](http://arxiv.org/abs/math/0603239). 


See Theorem 4.1 in my paper for what I think is the strongest result in this direction (as far as I know):



> 
> Gagola showed that the character of
>  $V$ vanishes on all but one nontrivial
>  conjugacy class if and only if there
>  exists a subgroup $N$ of $G$ such that
>  $N$ acts trivially on each simple
>  $\mathbb{C}[G]$-module except $V$.
> 
> 
> Let
>  $G$ be a group of order $n$ and $V$ a
>  simple $\mathbb{C}[G]$-module of
>  dimension $d$. Define $e$ such that
>  $n = d(d+e)$. Assume that there
>  exists a normal subgroup $N \neq
> > \{1\}$ such that $N$ acts trivially on
>  each simple $\mathbb{C}[G]$-module
>  other than $V$. Let $x$ be a
>  nontrivial element of $N$ and $C$ be
>  the centralizer of $x$ in $G$. Then
>  there exist a prime number $p$, a
>  positive integer $k$ and a
>  non-negative integer $m$ such that:
> 
> 
> * $N$ is elementary abelian of order $p^k$,
> * $C$ has order $p^k e^2$, and $d = e(p^k-1)$, and $n = e^2 p^k (p^k-1)$,
> * $C$ is a Sylow $p$-subgroup of $G$ and $e = p^m$,
> * if $H$ is any group such that $N \subseteq H \subseteq C$ and $|H| >
> > p^{k+m}$, then $N \subseteq [H,H]$.
> 
> 
> The above conditions are also sufficient, see Thm 4.3
> 
> 
> 


For n bigger than 2 there may be some results in MR1487363 Chillag "On zeroes of characters of finite groups."


