Is the integral closure of a local domain in a finite extension of its field of fractions semi-local? 
Is the integral closure of a local domain in a finite extension of its field of fractions semi-local?

If the answer is negative, I wonder under what conditions it would be semi-local.
EDIT
Here's an example of a local domain which is not necessarily a Japanese ring.
Let $A$ be a valuation ring, $K$ its field of fractions.
Let $P$ be the maximal ideal of $A$.
Let $L$ be a finite extension of $K$.
Let $B$ be the integral closure of $A$ in $L$.
It is well-known that there exist only finitely many valuation rings of $L$ dominating $A$.
Let $M$ be a maximal ideal of $B$.
It is well-known that $M$ lies over $P$.
Hence $B_M$ dominates $A$.
There exists a valuation ring $R$ of $L$ dominating $B_M$.
Since $M$ is determined by $R$ and $R$ dominates $A$, there exist only finitely many maximal ideals of $B$.
Hence $B$ is a semilocal ring.
Note that $B$ is not necessarily a finite $A$-module(even if $A$ is a discrete valuation ring).
Hence $A$ is not necessarily a Japanese ring.

Here are some sufficient conditions:
Let $A$ be a local domain, $K$ its fraction field,  $L$ a finite extension of $K$, and $B$ the integral closure of $A$ in $L$.  If $A$ is Noetherian and integrally closed, and $L$ is separable over $K$, then $B$ is necessarily finite over $A$, and so is semi-local (by going-up/down-type theorems).
If $A$ is not integrally closed, but its integral closure in $K$ is finite over itself, then again $B$ will be finite over $A$.  
The condition that the integral closure of $A$ in $K$ be finite over $A$ is (at least by some people) called N-1.  More generally, the condition that $B$ be finite over $A$ is called N-2, or Japanese.  (The letter N here stands for Nagata, and I believe Grothendieck coined the adjecive Japanese for these rings because these properties were studied by Nagata and the commutative algebra school around him in Japan.)
So if $A$ is a Japanese ring, then $B$ will be finite over $A$, and hence semilocal.  Of course, this is rather tautological: its utility follows from the fact that many rings (indeed, in some sense, most rings --- i.e. most of the rings that come up in algebraic number theory and algebraic geometry) are Japanese.  E.g. all finitely generated algebras over a field, or over $\mathbb Z$, or over a complete local ring, are Japanese.
Here are some useful wikipedia entries related to this topic: Nagata rings and Excellent rings.
