There are some theorems about various zeta functions which states the rationality of those. For example, when you consider Igusa's zeta function, roughly the generating series of solutions mod $p^n$ for $f\in\mathbb{Z}[x]$ -i.e. $P(T)=\sum \#\{x \ mod \ p^n ; f(x)=0 \}T^n$, it is rational. However, function $P$ is supposed to be more complicated when $f$ is more singular. It is an experience and it is what the proof of rationality somehow says.

My question is what are known methods to measure a complexity of rational functions or a singularity of varieties?

One can think about using a Galois something, lengths of blowups, norms... I'm really intrested what has been developed in this direction so far.