Equivalence problem for one-counter automata
I have no time to check the references in detail, but Ibarra states in http://dx.doi.org/10.1007/BF01744294 that, according to 
Greibachand Baker and Book, the universe problem (is the language $\Sigma^*$?) for non-deterministic one-counter machines with at most one counter reversal is undecidable. 

In addition to this, Ibarra considers even more restricted classes for which this problem is undecidable.

Hence, even a subproblem of equivalence is undecidable for a subclass of non-deterministic one-counter machines.