The magnetization distributions in the neighborhood of the corners of soft ferromagnetic geometries are both theoretically and experimentally investigated. We have previously introduced the wall cluster—a collection of domain walls having one intersection line, the so-called cluster knot, in common—as a novel concept to get a better grip on the complex process of domain-structure conversions. These wall clusters can be classified according to the positions of the cluster knots into the free, the edge, and the corner clusters. The edge and free clusters, whose cluster knots coincide with one single edge and with no any edge of the geometry, respectively, have been discussed elsewhere. Here the micromagnetic characteristics of a corner cluster in thin layers are discussed, while the systems approach to corner clusters, in which the cluster is treated as a system of walls, is left to Part II. Bear in mind that the torque equilibrium is dominated by the total Maxwell field, composed of the external and demagnetizing field near the domain walls and the edges of the geometry. From this starting point, it is shown that no domain wall is present in a region in which the total field is not zero. Similarly, it is demonstrated that no line space-charge-density singularities can appear at a cluster knot. As a consequence, the magnetization in the domains can be divided into so-called rotation segments and uniform sectors, in which the magnetization rotates circularly around the cluster knot or is quasiuniform, respectively. Space-charge-density singularities occur at the bounding surfaces of the geometry. This surface-charge density generates a singularity in the total field in case this density does not reduce to zero in the corner. No domain wall can be present in the corner in these circumstances, so that the cluster knot is either shoved along one of the bounding edges of the corner or it disengages itself from the edges and the corner cluster is tranformed into a free cluster. It is experimentally demonstrated that these singularities in the field often, but not always, occur in the corner when an external field is applied. These and other above-mentioned statements are also supported by experimental evidence.