It is well known that the polynomial ring in $n$ variables is a free module over the non-constant symmetric functions and the quotient is isomorphic as an $S_n$ module to the left regular representation. The study of this quotient is classical, and among its well-known bases are the Artin monomials and the Schubert polynomials. Within the last ten years, Chalykh, Etingof, Felder, Feigin, Ginsburg and Veselov have developed a generalization of the notion of the invariants of a reflection group $G$. In particular, they defined a nested sequence of rings $Q_m$ indexed by non-negative integers such that $Q_0$ is the whole polynomial ring and the limit, $Q_\infinity$, is the usual ring of $G$-invariants. They called these rings the $m$-quasiivariants of $G$. Remarkably, it was shown that, for all $m$, the ring $Q_m$ is a free module over the non-constant invariants of $G$, and the quotient is isomorphic to the left regular representation. However, even in the case of $G = S_n$, a basis for this quotient is unknown. In this talk I will present some recent progress, joint with Gregg Musiker, on the problem of finding such a basis.