For the first time, the Cluster spacecraft have collected 3-D information on magnetic field structures at small to medium scales in the Earth's dayside magnetosphere. We focus here on the first application of the Curlometer (direct estimation of the electric current density from curl(B), using measured spatial gradients of the magnetic field) analysis technique. The applicability of this multipoint technique is tested, for selected events within the data set, in the context of various mission constraints (such as position, timing, and experimental accuracy). For the Curlometer, nonconstant spatial gradients over the spacecraft volume, time dependence, and measurement errors can degrade the quality of the estimate. The estimated divergence of the magnetic field can be used to monitor (indirectly) the effect of nonconstant gradients in the case of many magnetic field structures. For others, and at highly distorted spacecraft configurations, this test may not reflect the quality of the Curlometer well. The relative scales and relative geometry between the spacecraft array and the structures present, as well as measurement errors, all are critical to the quality of the calculation. We demonstrate that even when instrumental and other errors are known to contribute to the uncertainty in the estimate of the current, a number of current signatures within the magnetosphere can be plausibly determined in direction, if not absolute size. A number of examples show consistent currents at the magnetopause, both separate from, and nearby or in the cusp region. Field-aligned currents near the polar cap boundary are also estimated reliably. We also demonstrate one example of an anomalous current arising from the effect of a highly distorted spacecraft configuration.