We present a comprehensive spectral and morphological analysis of the remnant of Supernova (SN) 1987A with the Australia Telescope Compact Array (ATCA) and the Atacama Large Millimeter/submillimeter Array (ALMA). The non-thermal and thermal components of the radio emission are investigated in images from 94 to 672 GHz ($λ$ 3.2 mm to 450 $μ$m), with the assistance of a high-resolution 44 GHz synchrotron template from the ATCA, and a dust template from ALMA observations at 672 GHz. An analysis of the emission distribution over the equatorial ring in images from 44 to 345 GHz highlights a gradual decrease of the east-to-west asymmetry ratio with frequency. We attribute this to the shorter synchrotron lifetime at high frequencies. Across the transition from radio to far infrared, both the synchrotron/dust-subtracted images and the spectral energy distribution (SED) suggest additional emission beside the main synchrotron component ($S_{ν}∝ν^{-0.73}$) and the thermal component originating from dust grains at $T∼22$ K. This excess could be due to free-free flux or emission from grains of colder dust. However, a second flat-spectrum synchrotron component appears to better fit the SED, implying that the emission could be attributed to a pulsar wind nebula (PWN). The residual emission is mainly localised west of the SN site, as the spectral analysis yields $-0.4\lesssimα\lesssim-0.1$ across the western regions, with $α∼0$ around the central region. If there is a PWN in the remnant interior, these data suggest that the pulsar may be offset westward from the SN position. ; Comment: ApJ accepted. 21 pages, emulateapj. References updated