Many objects from microparticles to insects can climb curved liquid menisci by distorting the liquid interface around them. Here we show how the wrinkly deformations within a flexible sheet can propel it along a curved fluid interface. We measure the energy landscape of a circular polymer film on an axisymmetric meniscus. At small interface curvatures, we observe dynamics consistent with previous theory for a stretching-dominated regime. At large interface curvatures, we uncover a new dynamical regime when the sheet is wrinkled. These results are captured by a geometric model for an inextensible sheet that is free to compress via wrinkling. Experiments with curved shells show that they too seek regions that match their intrinsic curvature, which could be useful for sensing curvatures or delivering cargo to precise locations.