Superconducting YBa2Cu3Oy (YBCO) foams were prepared using commercial open-cell, polyurethane foams as starting material to form ceramic Y2BaCuO5 foams which are then converted into superconducting YBCO by using the infiltration growth process. For modelling the superconducting and mechanical properties of the foam samples, a Kelvin-type cell may be employed as a first approach as reported in the literature for pure polyurethane foams. The results of a first modelling attempt in this direction are presented concerning an estimation of the possible trapped fields (TFs) and are compared to experimental results at 77 K. This simple modelling revealed already useful information concerning the best suited foam structure to realize large TF values, but it also became obvious that for various other parameters like magnetostriction, mechanical strength, percolative current flow and the details of the TF distribution, a refined model of a superconducting foam sample incorporating the real sample structure must be considered. Thus, a proper description of the specific microstructure of the superconducting YBCO foams is required. To obtain a set of reliable data, YBCO foam samples were investigated using optical microscopy, scanning electron microscopy and electron backscatter diffraction (EBSD). A variety of parameters including the size and shape of the cells and windows, the length and shape of the foam struts or ligaments and the respective intersection angles were determined to better describe the real foam structure. The investigation of the foam microstructures revealed not only the differences to the original polymer foams used as base material, but also provided further insights to the infiltration growth process via the large amount of internal surface in a foam sample.