A directed graph is called Eulerian if it has an Eulerian circuit, which is equivalent to requiring that the kernel of its Laplacian matrix is minimal. We call a directed graph coEulerian if the cokernel of its Laplacian matrix is maximal. While it is easy to see that a random graph is almost never Eulerian, calculating the probability that it is coEulerian is considerably harder. We sketch a proof that the limit of this probability is bounded by a constant around 0.43, and talk about the obstacles to proving that the limit is exactly this constant. In the last part of the talk, we talk about results and conjectures on the probability that a random rectangular matrix is surjective, and their implications for random graph theory.