We will consider some Hardy-type inequalities for the Dirac operator $\mathcal D$ in 3d: they are a very useful tool to construct self-adjoint extensions for $\mathcal D + V$, where $V$ is a general Hermitian matrix valued potential $V$, such that $|V(x)| \sim 1/|x|$, and they have been exploited to prove the validity of min-max formulas determining the eigenvalues of $\mathcal D + V$ in the spectral gap. In a recent work in collaboration with Fabio Pizzichillo and Luis Vega (Basque Center for Applied Mathematics, Bilbao) we prove a sharp Hardy-type inequality for the Dirac operator and we exploit this to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix valued potentials $V$ such that $\sup_x |x||V(x)| <1$: we characterise its eigenvalues in terms of the Birman--Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if $V$ verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that $V$ is the Coulomb potential. We will describe these results, and finally we will discuss some possibile generalisations to the case that $\sup_x |x||V(x)| =1$.