A rather nice example is the monoid of $3\times 3$ integer matrices.
Its membership problem is unsolvable, indeed so is the problem when $x$
is restricted to be the zero matrix. This is another way to state the
unsolvability of the mortality problem for $3\times 3$ integer matrices,
previously mentioned in [this MO answer][1].

[1]:http://mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/15766#15766