  In this paper we explore finite rank perturbations of unilateral weighted
shifts $W_\alpha$. First, we prove that the subnormality of $W_\alpha$ is never
stable under nonzero finite rank pertrubations unless the perturbation occurs
at the zeroth weight. Second, we establish that 2-hyponormality implies
positive quadratic hyponormality, in the sense that the Maclaurin coefficients
of $D_n(s):=\text{det} P_n [(W_\alpha+sW_\alpha^2)^*, W_\alpha+s W_\alpha^2]
P_n$ are nonnegative, for every $n\ge 0$, where $P_n$ denotes the orthogonal
projection onto the basis vectors $\{e_0,...,e_n\}$. Finally, for $\alpha$
strictly increasing and $W_\alpha$ 2-hyponormal, we show that for a small
finite-rank perturbation $\alpha^\prime$ of $\alpha$, the shift
$W_{\alpha^\prime}$ remains quadratically hyponormal.
