To my mind the sexiest of open problems in probability is to show that there is "no percolation at the critical point" (mentioned in particular in section 4.1 of Gordon Slade's contribution to the Princeton Companion to Mathematics). A capsule summary: write $\mathbb{Z}_{d,p}$ for the random subgraph of the nearest-neighbour $d$-dimensional integer lattice, obtained by independently keeping each edge with probability $p$. Then it is known that there exists a critical probability $p_c(d)$ (the percolation threshold}) such that for $p < p_c$, with probability one $\mathbb{Z}_{d,p}$ contains no infinite component, and for $p > p_c$, with probability one there exists an unique infinite component.
The conjecture is that with probability one, $\mathbb{Z}_{d,p_c(d)}$ contains no infinite component. The conjecture is known to be true when $d =2$ or $d \geq 19$.