As explained in  [Singularities of pairs](https://mathoverflow.net/questions/42010/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?



Dimension and singularities of the minimal log canonical center
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
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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
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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
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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).


