  We prove that almost every level set of a Sobolev function in a planar domain
consists of points, Jordan curves, or homeomorphic copies of an interval. For
monotone Sobolev functions in the plane we have the stronger conclusion that
almost every level set is an embedded $1$-dimensional topological submanifold
of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and
minimum of the function in an open set are attained at the boundary. Our result
is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function
in a planar domain almost every value is a regular value. As an application we
show that monotone Sobolev functions in planar domains can be approximated
uniformly and in the Sobolev norm by smooth monotone functions.
