  We establish zero-one laws and convergence laws for monadic second-order
logic (MSO) (and, a fortiori, first-order logic) on a number of interesting
graph classes. In particular, we show that MSO obeys a zero-one law on the
class of connected planar graphs, the class of connected graphs of tree-width
at most $k$ and the class of connected graphs excluding the $k$-clique as a
minor. In each of these cases, dropping the connectivity requirement leads to a
class where the zero-one law fails but a convergence law for MSO still holds.
