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Scanning transmission x-ray microscopy has been used to image electric currents in superconducting films at temperatures down to 20 K. We detect significant deviations from a regular current path driven by macroscopic geometrical constraints. The magnetic stray field of supercurrents in a thin YBaCuO film is mapped into a soft-magnetic coating of permalloy. The so-created local magnetization of the ferromagnetic film can be detected by dichroic absorption of polarized x rays. To enable high-quality measurements in transmission geometry, the whole heterostructure of ferromagnet, superconductor, and single-crystalline substrate has been thinned to an overall thickness of less than 1 \textmu{}m. With this technique, local supercurrents can be analyzed in a wide range of temperatures and magnetic fields. The less than 100 nm spatial resolution of the magnetic signal together with simultaneously obtained nanostructural data allow the correlation of local supercurrents with the micro- and nanostructure of the superconducting film.