Here are two from ergodic theory:
(The problem of smooth realizations) Let $X$ be a Lebesgue space with measure $\mu$, and let $T:X\to X$ be a transformation preserving the measure $\mu$. If the entropy $h_\mu(T)$ is finite, is $(X,T,\mu)$ always measurably isomorphic to a smooth system $(M,f,v)$, where $M$ is a compact manifold, $f$ is a diffeomorphism of $M$ and $v$ is a smooth volume?
(Furstenberg's $\times 2 \times 3$ problem) Does there exist a Borel probability measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$, which is neither discrete nor Haar measure, and which is invariant under both $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$?
For the first problem, as far as I know there has been no significant progress.
For Furstenberg's conjecture, Furstenberg himself solved the analog question for sets (answer is negative), and Rudolph proved that the answer is negative under an extra positive entropy assumption. While there has been a huge amount of progress in the positive entropy case since, the zero entropy case remains untractable despite the simplicity of the statement.
