A classical question in analytic number theory is: given a polynomial with integer coefficients, how often does it take squarefree values? In arithmetic statistics, we are particularly interested in the case of discriminant polynomials. In this talk, I will present several different cases of this question. First, we will consider a classical result of Davenport--Heilbronn which considers the case of discriminants of binary cubic forms. Motivated by this case, I will discuss two works, joint with Manjul Bhargava and Xiaoheng Wang, in which we consider the spaces of degree-n polynomials and degree-n binary forms. We will prove that a positive proportion of these polynomials and forms have squarefree discriminant.