This paper examines the possible occurrence of tearing modes in the dayside magnetopause. First, the expected magnetic signature of tearing, as obtained from existing theory, is reviewed. Magnetometer data from one terrestrial magnetopause crossing and one crossing of Jupiter's magnetopause are then examined in detail. Magnetic field oscillations are found in three subsegments of the terrestrial crossing at a frequency of 0.1-0.2 Hz and with peak amplitudes of 5-10 nonotesla (nT), and in one segment of the Jovian crossing, at 0.05-0.1 Hz and with 2-nT amplitude. The frequency range, as well as the orientation of the magnetic field perturbation vectors, agrees with a model in which tearing-produced magnetic islands are convected past the satellite with the plasma in the current layer. In both cases the magnetopause structure was of the rotational discontinuity type with a nonvanishing normal magnetic field component. Hence, if the tearing structures were active, i.e., growing, at the observation site, ion tearing (or perhaps resistive tearing, with the resistivity provided by microturbulence) must be involved. But it is also possible that the structures were passive, consisting of 'debris' from active tearing elsewhere on the magnetopause surface, this debris being convected along the magnetopause past the observation site.