Why don't dispersions like median deviation and mode deviation exists on the lines of mean deviation? Or to say in the formula MD=$\frac{1}{N}\sum_{i=1}^{N}|x_i-\overline{x}|$ why can't we have median or mode instead of mean($\overline{x}$) and as a result speak about additional possible measures of deviation?
 A: Yes we can, there is, for example, median absolute deviation (MAD):
$$ \operatorname{MAD}(X) = \operatorname{median}\left(\ \left| X_{i} - \operatorname{median} (X) \right|\ \right)$$
It has even some nice properties, e.g. $1.4826 \times \operatorname{MAD}(X)$ is estimator for standard deviation of normally distributed $X$. You can read more about it in the paper by Rousseeuw and Croux (1993).
We often look at much more then mean of errors, e.g. quantiles of errors, like in standard output of summary(lm) in R:
Call:
lm(formula = mpg ~ ., data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4506 -1.6044 -0.1196  1.2193  4.6271 

We also look at distributions of errors by plotting residuals and absolute errors etc. 
It is not that commonly used because mean is more sensitive to outliers then median, and in most cases this is a desired property when talking about error measures.

Rousseeuw, P.J. and Croux, C. (1993). Alternatives to the median absolute deviation. Journal of the American Statistical Association. 88 (424): 1273–1283. 
