Methane is the simplest molecule I know of that takes the shape of a platonic solid — in this case, the tetrahedron. Wikipedia has a page on platonic hydrocarbons that includes molecules with tetrahedron, cube, and dodecahedron shapes. There are lots of molecules that have octahedral geometry (e.g. sulfur hexafluoride).

Are there any molecules that form an icosahedron? The closest I've found is $\ce{C_60}$, buckminsterfullerene, which takes the form of an Archimedean solid, the truncated icosahedron.