In this talk we'll consider polynomial vector fields on $\mathbb{C}^2$ with non-degenerate singularities. At each singular point, the linearizion matrix of the vector field has two non-zero eigenvalues (the spectrum of the matrix). The spectra of singularities is the collection of all such eigenvalues.

We aim to understand the spectra of singularities of generic polynomial vector fields of a fixed degree. These are some questions we care about: are the spectra at each of the singularities independent to one another? If not, which relations are satisfied among them? Does the spectra of singularities completely determine the vector field? Which collections of numbers are realizable as the spectra of a polynomial vector field?

We will discuss these questions and answer them for the case of quadratic vector fields. At the end, we'll present more results and more questions that come up when we extend the vector fields to singular foliations on $\mathbb{C}\mathrm{P}^2$ and take into account the Baum-Bott indices of the singularities that appear "at infinity".