Dimension and singularities of the minimal log canonical center As explained in  Singularities of pairs by  Karl Schwede, log canonical center is quite useful for induction. For example, to study effective freeness by induction on dimension of log canonical centers.  In general, log canonical centers are not smooth. 
Question:
How does the dimension and multiplicity of a log canonical center related to the pair itself?
Let $(X, D)$ be a log canonical pair and $Z$ the minimal log canonical center of the pair? What property of $(X, D)$ will control the dimension of $Z$?  How singular could $Z$ be?
 A: If $(X, D)$ is a log canonical pair and $Z$ is the minimal log canonical center, then for some appropriate $D_Z$, the pair $(Z, D_Z)$ is KLT.  In particular, $Z$ always has rational singularities.  Thus I'd say the singularities of minimal LC centers are not so bad. See the papers of Kawamata on subadjunction.  Florin Ambro also has a couple surveys on the arXiv.
Dimension
In terms of the dimension, without more information, the dimension of $Z$ can be anything from $\dim X - 1$-dimensional (if and only if $(X, D)$ is PLT) to $0$ dimensional.  Certainly if $(X, D)$ is Kawamata log terminal on an open set $U$, then $Z \subseteq X \setminus U$.  So you can control dimension in that way.
Multiplicity
If $X$ is smooth of dimension $n$, Stefan Helmke proved that any union of $d$-dimensional log canonical centers has multiplicity $\leq {n \choose d}$.  He actually has a number of more precise results as well, see his paper in Duke for more details.
Applications
In many applications, say for the construction of global sections of adjoint divisors, being able to construct 0-dimensional centers would make things very easy (see for Example chapter 10 of Rob Lazarsfeld's book Positivity in algebraic geometry, or the papers of Kawamata on Fujita's conjecture).  On the other hand, people have certainly looked for $\dim X - 1$-dimensional centers as well (I believe there is work of Mella, Ambro and Alexeev on this, look up Ladders on Fano varieties).
A: This is an addition to Karl's answer:
A recent paper by Kollár introduces new powerful techniques to handle lc centers. He also proves some very interesting properties, in particular kind of an extension of adjunction to higher codimension lc centers. The paper is here.
Another fact about lc centers is that an arbitrary union of lc centers has Du Bois singularities by this paper.
