We report that a polymer molecule can be trapped spatially and conformationally using a microflow that has at least two stagnation points (or two points with equal velocity) and a net flow orthogonal to the line connecting them. Examples include a Taylor vortex flow and an electro-osmotic flow in a channel with surfaces that have a sinusoidal charge. Simulating the motion of a polymer molecule in these flows using Brownian dynamics, we find that such flows produce a curved polymer conformation, leading to an elastic force that drives migration against the flow, thus stabilizing this conformation. Simulations with hydrodynamic interactions confirm these predictions and show that there exists a repulsive interaction between two trapped polymers.