Nice question.  I am also keen to learn of more examples.  

Here are some more details on the Hamiltonian cycle example.  This is expressible in $\mathsf{MSO_2}$ since testing if a graph is connected is expressible in $\mathsf{MSO}_2$ and testing if a set of edges induces a $2$-regular subgraph is clearly expressible in $\mathsf{MSO}_2$.  Indeed, connectivity is actually expressible in $\mathsf{MSO_1}$ since a graph $G$ is connected if and only if for all partitions $A \cup B$ of $V(G)$ there is an edge between $A$ and $B$.  However, $\mathsf{MSO}_1$ does not know the difference between balanced complete bipartite graphs (which do have Hamiltonian cycles) and unbalanced complete bipartite graphs (which do not have Hamiltonian cycles).