Abstract: If the location of a singularity of an ODE varies as the initial conditions change, then the singularity is said to be movable. This is on contrast to fixed singularities, which only occur at points where the equation is singular in some sense. Movable singularities in the complex domain play a special role in the theory of integrable systems. In this talk it will be shown that for several classes of second-order ordinary differential equations, the only movable singularities that can be reached by analytic continuation along finite-length curves are poles or algebraic branch points. Although these solutions have simple local singularity structure, globally they can be very complicated. We use Nevanlinna theory to classify some equations that admit solutions that are algebraic over the field of meromorphic functions.