The set of polynomials of degree 4 with all roots on the unit circle, and inscribed easily described regions.
Large resultants of families of polynomials generated as in Dobrowolski’s Lemma (which leads to the best known lower bound for Mahler measure as a function of degree).
The eigenvalues of a real random asymmetric matrix with iid normal entries.
I’m not sure what this is, but it’s interesting.
The zeros of Wronskians of consecutive Hermite (and other classical orthogonal) polynomials are pretty wild.
Pair correlation in the scaling limit near the real edge in Ginibre’s real ensemble.