Summary: Some draining or coating fluid-flow problems, in which surface tension forces are important, can be described by third-order ordinary differential equations. Accurate computations are provided here for examples such as \(y\prime''(x)=-1+1/y^ 2\) that permit the boundary condition \(y\to 1\) as \(x\to -\infty\), so modelling a layer of fluid that is asymptotically uniform behind the draining front. The ultimate fate of the solution as x increases is studied for the above example, and for a generalisation involving a small parameter \(\delta\) such that this example is recovered in the limit as \(\delta\to 0\), but which is such that \(y\to \delta\) as \(x\to +\infty\), so modelling draining over an already-wet surface. Matched asymptotic expansions are then used to derive limiting results for small \(\delta\), this being a singular perturbation since the problem with \(\delta =0\) does not permit \(y=0\). The physical basis for this singularity is the well-known impossibility of moving a contact line over a dry nonslip surface. Other modifications that avoid the singularity by allowing slip are also discussed.