Let $\pi : C_g \to M_g$ be the universal curve over the moduli stack of genus $g$ curves. Let $\omega_\pi$ be the relative canonical bundle. Then $\mathbb{H} := \pi_\ast \omega_\pi$ is a rank $g$ vector bundle, and it is called the Hodge bundle. Its $i$-th Chern class $c_i(\mathbb{H})$ is denoted by $\lambda_i$. These classes are called Hodge classes.

I'd like to know: What are the (known) polynomial relations among the $\lambda_i$'s? Is there an exhaustive list?

A simple basic relation is $\lambda_1^2 = 2\lambda_2$. I can derive this as follows:

By Grothendieck-Riemann-Roch, we have $ch(\pi_! \omega_\pi) = ch(\mathbb{H})-1 = \pi_\ast ( ch(\omega_\pi) \cdot td(\omega_\pi^\vee))$. By a simple calculation, the degree 3 (if you use Chow groups) (degree 6 if you use cohomology) term of $ch(\omega_\pi) \cdot td(\omega_\pi^\vee)$ is zero. Thus the degree 2 term $ch_2(\mathbb{H})$ of the left hand side is zero. We also have $c_2(\mathbb{H}) = \frac{ch_1(\mathbb{H})^2}{2} - ch_2(\mathbb{H})$, hence we conclude $2c_2(\mathbb{H}) = ch_1(\mathbb{H})^2 = c_1(\mathbb{H})^2$.