Can one obtain surfaces with interesting invariants as resolutions of singular surfaces? (Perhaps a not very well defined question)
Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities.  As the simplest case consider a degenerating family of surfaces in $\Bbb P^3$. Let $\tilde{S}\to S_0$ be the minimal resolution of singularities.


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*Can one obtain in this way some interesting surfaces (in the sense of geography problem)?
(References on works/summaries?)

*Suppose I have some inequality among the local singularity invariants of $S_0$. The inequality will translate into some inequality among the invariants of $\tilde{S}$. Are there examples when such singularity inequalities are useful?
 A: The answer to your first question is definitely yes. In fact, many interesting examples of smooth, complex algebraic surfaces are given by desingularization of singular ones. The subject is too broad to be fully treated in a MO post. However, let me just recall the paper  by Stephan Endraß, Ulf Persson and Jan Stevens
"Surfaces with triple points", http://arxiv.org/abs/math/0010163
in which the authors consider the effect of imposing a finite number of ordinary triple points to the invariants of a surface in $\mathbb{P}^3$. In fact, in contrast to the case of ordinary double points (classified by Du Val as those which "do not effect the conditions of adjunction"), when we impose one or more ordinary triple points both the invariants of the surface and its type in the classification may change.
The purpose of the paper mentioned above is to point out the intricate geometry of examples with many triple points, and how it fits with the general classification of surfaces. 
A: Recently many interesting complex surfaces of general type with $p_g=q=0$ and small (topological, algberaic) fundamental groups are constructed via the so-called $\mathbb{Q}$-Gorenstein smoothing method. You may refer the paper [Invent. Math. 2007] by Yongnam Lee and Jongil Park.
