In response to this CompSciTheory question,
[A simple problem whose decidability is not known
](https://cstheory.stackexchange.com/q/18846/337),
I [posted](https://cstheory.stackexchange.com/a/32797/337) that:

> It is unknown whether or not
[it is decidable to determine if a given shape can tile the plane](https://cstheory.stackexchange.com/q/32538/337),

referring to an earlier cstheory question.
This is even open for polyomino tiles.