Most linear experimental design problems assume homogeneous variance although heteroskedastic noise is present in many realistic settings. Let a learner have access to a finite set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$ that can be probed to receive noisy linear responses of the form $y=x^{\top}θ^{\ast}+η$. Here $θ^{\ast}\in \mathbb{R}^d$ is an unknown parameter vector, and $η$ is independent mean-zero $σ_x^2$-sub-Gaussian noise defined by a flexible heteroskedastic variance model, $σ_x^2 = x^{\top}Σ^{\ast}x$. Assuming that $Σ^{\ast}\in \mathbb{R}^{d\times d}$ is an unknown matrix, we propose, analyze and empirically evaluate a novel design for uniformly bounding estimation error of the variance parameters, $σ_x^2$. We demonstrate the benefits of this method with two adaptive experimental design problems under heteroskedastic noise, fixed confidence transductive best-arm identification and level-set identification and prove the first instance-dependent lower bounds in these settings. Lastly, we construct near-optimal algorithms and demonstrate the large improvements in sample complexity gained from accounting for heteroskedastic variance in these designs empirically.