A fact that has been recently studied is the stable irrationality of $X$, namely the fact that the product $X\times \mathbf{P}^n$ is not rational for any $n$. This, as you might know, implies irrationality as well. You can find out more in the works of Colliot-Thélène and Pirutka; a good survey is, for instance, http://cims.nyu.edu/~pirutka/survey.pdf

The main arguments used involve unramified cohomology and Brauer groups. Basically, one knows that the unramified cohomology groups $H_\mathrm{rm}^2 (k(X)/k,\mathbf{Z}/2)\simeq \mathrm{Br}(X)[2]$ are related to stable rationality. In particular, if $\mathrm{Br}(X)[2]$ does not vanish, then $X$ is not stably rational.

In the above survey paper a formula for the unramified cohomology of conic bundles of the form $\pi : X\longrightarrow \mathbf{P}^2$ is given, attributed to Colliot-Thélène. This formula employs geometric conditions on the discriminant locus to determine the behaviour of $\mathrm{Br}(X)[2]$.

A similar formula, but for conic bundles over threefolds with some "quasi-rational" conditions, was given in the paper https://arxiv.org/abs/1610.04995 by Auel, Boehning, von Bothmer and Pirutka. The formula addresses the unramified cohomology groups of conic bundles of the form $\pi : X\longrightarrow B$, where $B$ is a smooth projective 3-fold with $H^3_{\mathrm{ét}}(B,\mathbf{Z}/2)=0$ and $\mathrm{Br}(B)[2]=0$, and again uses heavily the discriminant locus to describe its behaviour.