We initiate a study of the quasisymmetric uniformization of naturally arising random fractals and show that many of them fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with the trace, the graph of Brownian motion, and various variants of the Schramm-Loewner evolution $\mathrm{SLE}_κ$ for $κ>0$, and show that a.s. neither is a quasiarc. After that, we study the conformal loop ensemble $\mathrm{CLE}_κ$, $κ\in (\frac{8}{3}, 4]$, and show that the collection of all points outside the loops is a.s. homeomorphic to the standard Sierpiński carpet, but not quasisymmetrically equivalent to a round carpet.